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# Long time justification of wave turbulence theory

Yu Deng  Department of Mathematics, University of Southern California, Los
Angeles, CA, USA<EMAIL_ADDRESS>and  Zaher Hani  Department of Mathematics,
University of Michigan, Ann Arbor, MI, USA<EMAIL_ADDRESS>
###### Abstract.

In a series of previous works [15, 16, 18], we gave a rigorous derivation of
the homogeneous wave kinetic equation (WKE) up to small multiples of the
_kinetic timescale_ , which corresponds to short time solutions to the wave
kinetic equation. In this work, we extend this justification to arbitrarily
long times that cover the full lifespan of the WKE. This is the first long-
time derivation ever obtained in any nonlinear (particle or wave) collisional
kinetic limit.

###### Contents

  1. 1 Introduction
    1. 1.1 Setup and the main result
      1. 1.1.1 The wave kinetic equation
      2. 1.1.2 The main result
    2. 1.2 Background and history
    3. 1.3 The difficulty
    4. 1.4 The strategy
    5. 1.5 Future horizons
      1. 1.5.1 Extension of Lanford’s theorem
      2. 1.5.2 Fluctuations and large deviations
      3. 1.5.3 Diffusion and other asymptotic limits
  2. 2 Description of main ideas
    1. 2.1 An example
    2. 2.2 Building up gardens: A naive attempt
    3. 2.3 Canonical layering and the true ansatz
      1. 2.3.1 Main components of the proof
      2. 2.3.2 Structure of the paper
  3. 3 Preparations for the proof
    1. 3.1 First reductions
    2. 3.2 Moments and cumulants
    3. 3.3 Norms, parameters, and notations
      1. 3.3.1 Norms
      2. 3.3.2 Parameters
      3. 3.3.3 Notations
    4. 3.4 Combinatorial structures
      1. 3.4.1 Trees, gardens and couples
      2. 3.4.2 Layerings
    5. 3.5 The inductive scheme
    6. 3.6 The Duhamel expansion
    7. 3.7 Proof of Proposition 3.9 and Theorem 1.1
  4. 4 Layered regular objects I: Combinatorics
    1. 4.1 Regular objects and structure theorem
    2. 4.2 Layered regular objects
  5. 5 Layered regular objects II: Decay for non-coherent objects
    1. 5.1 The non-coherent estimates
  6. 6 Layered regular objects III: Asymptotics for coherent objects
    1. 6.1 The coherent estimates
    2. 6.2 Preparations for the proof
    3. 6.3 Proof of Propositions 6.1–6.3
  7. 7 Molecules, vines and twists
    1. 7.1 Molecules
    2. 7.2 Blocks, vines and ladders
      1. 7.2.1 Blocks
      2. 7.2.2 Vines and ladders
    3. 7.3 Twists
    4. 7.4 Layered full twists
  8. 8 Layered vines and ladders: Combinatorics
    1. 8.1 Number of coherent layerings
  9. 9 Analysis of vines: Role of cancellation and incoherency
    1. 9.1 A general estimate for exponential sums
      1. 9.1.1 Number theory lemmas
      2. 9.1.2 Exponential sums
    2. 9.2 Application to vine chains
      1. 9.2.1 Reduction and twisting of $\mathcal{K}_{\mathcal{G}}$
      2. 9.2.2 (CL) vine chains
  10. 10 Stage 1 reduction
    1. 10.1 Vine collections $\mathscr{V}$ and $\mathscr{V}_{0}$
    2. 10.2 Reduction to $\mathcal{G}_{\mathrm{sb}}$ and $\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$
    3. 10.3 Analysis of ladders: $L^{1}$ bound and extra decay
    4. 10.4 Properties of $\mathbb{M}_{\mathrm{sb}}$
  11. 11 Stage 2 reduction and proof of Proposition 3.11
    1. 11.1 Reduction to counting estimates
      1. 11.1.1 Ladders and $L^{1}$ bounds
    2. 11.2 Reduction to large gap molecules
      1. 11.2.1 Cutting operations and estimates
    3. 11.3 The rigidity theorem
      1. 11.3.1 Proof of Proposition 3.11
  12. 12 Linearization and proof of Proposition 3.12
    1. 12.1 Flower trees and construction of parametrix
    2. 12.2 Proof of Proposition 3.12
  13. A Miscellaneous results
    1. A.1 Cumulants
    2. A.2 Analytical lemmas
    3. A.3 Combinatorial lemmas
    4. A.4 Counting lemmas
  14. B Failure of a physical space $L^{1}$ bound
  15. C Table of notations

##  1\. Introduction

The problem of rigorously justifying the laws of kinetic theory and more
generally statistical physics, starting from Hamiltonian first principles, was
first raised to prominence by Hilbert in his famous list of problems announced
at the ICM in 1900.

More than a century has passed since Hilbert announced this problem—now called
_Hilbert’s Sixth Problem_ —and during this time, a tremendous amount of
progress has been achieved towards its end. We shall review this progress more
thoroughly in Section 1.2, but it can be summarized briefly as follows: In the
classical context of particle kinetic theory, the turning point was Lanford’s
result [47] in 1975 which justified Boltzmann’s kinetic theory up to small
kinetic times; this result was later completed and developed upon in many
works, most notably [30]. In the context of wave kinetic theory (also known as
wave turbulence theory), where colliding particles are replaced by nonlinearly
interacting waves, the parallel results were only recently achieved by the
authors [15, 16, 18] following several important developments after the turn
of the century in linear or equilibrium settings. However, for all this time,
one of the biggest challenges in kinetic theory remained open, which is to
resolve Hilbert’s sixth problem for a nonlinear system (such as classical
Boltzmann or wave turbulence) over arbitrarily long time intervals in the
kinetic timescale.

In this paper, we give the _first ever long-time justification result_ of a
kinetic limit for a nonlinear Hamiltonian (particle or wave) collisional
system: we derive the homogeneous wave kinetic equation (WKE), from the cubic
nonlinear Schrödinger (NLS) equation, up to _large_ multiples of the kinetic
time $T_{\mathrm{kin}}$, that cover the _full lifespan_ of the WKE. We also
obtain associated results such as propagation of chaos, density evolution, and
the hierarchical evolution of higher moments over the same long time range.
This extends the results of [15, 16, 17, 18], which concern only sufficiently
small multiples of $T_{\mathrm{kin}}$.

###  1.1. Setup and the main result

We start by describing the setup and main result of this paper, before moving
to the background and discussions in Section 1.2. Following [18], in dimension
$d\geq 3$, consider the cubic nonlinear Schrödinger equation

$\left\\{\begin{split}&(i\partial_{t}-\Delta)u+\alpha|u|^{2}u=0,\quad
x\in\mathbb{T}_{L}^{d}=[0,L]^{d},\\\
&u(0,x)=u_{\mathrm{in}}(x)\end{split}\right.$ (NLS)

on the square torus111All results and proofs extend without change to
arbitrary rectangular tori as long as $0<\gamma<1$ (see below). For $\gamma=1$
we need a genericity condition on the torus. $\mathbb{T}_{L}^{d}=[0,L]^{d}$ of
size $L$. Here $\alpha$ is a parameter indicating the strength of the
nonlinearity, and

$\Delta:=\frac{1}{2\pi}(\partial_{x_{1}}^{2}+\cdots+\partial_{x_{d}}^{2})$

is the normalized Laplacian. Let $\mathbb{Z}_{L}^{d}:=(L^{-1}\mathbb{Z})^{d}$
be the dual of $\mathbb{T}_{L}^{d}$, and fix the space Fourier transform as

$\widehat{u}(t,k)=\frac{1}{L^{d/2}}\int_{\mathbb{T}^{d}_{L}}u(t,x)e^{-2\pi
ik\cdot x}\,dx,\qquad
u(t,x)=\frac{1}{L^{d/2}}\sum_{k\in\mathbb{Z}_{L}^{d}}\widehat{u}(t,k)e^{2\pi
ik\cdot x}.$ (1.1)

Assume the initial data of (NLS) is given by

$u_{\mathrm{in}}(x)=\frac{1}{L^{d/2}}\sum_{k\in\mathbb{Z}_{L}^{d}}\widehat{u_{\mathrm{in}}}(k)e^{2\pi
ik\cdot
x},\quad\widehat{u_{\mathrm{in}}}(k)=\sqrt{\varphi_{\mathrm{in}}(k)}\cdot\mathfrak{g}_{k},$
(DAT)

where $\varphi_{\mathrm{in}}:\mathbb{R}^{d}\to[0,\infty)$ is a given Schwartz
function, and $\\{\mathfrak{g}_{k}\\}$ is a collection of i.i.d. random
variables. For concreteness, we will assume each $\mathfrak{g}_{k}$ is a
standard normalized Gaussian. The results remain true with suitable
modifications in non-Gaussian cases, see Remark 1.5 and [16].

Define the _kinetic (or Van Hove) time_

$T_{\mathrm{kin}}:=\frac{1}{2\alpha^{2}}.$

In this paper we will study the dynamics of (NLS) under the limit
$L\to\infty,\alpha\to 0$. We assume that they are related by
$\alpha=L^{-\gamma}$, so in particular
$T_{\mathrm{kin}}=\frac{1}{2}L^{2\gamma}$, where $\gamma\in(0,1]$ is a fixed
value called the _scaling law_ between $L$ and $\alpha$. This is the full
range of admissible scaling laws for the problem as we explain in Remark 1.4
below.

####  1.1.1. The wave kinetic equation

The wave kinetic equation is given by:

$\left\\{\begin{split}&\partial_{\tau}\varphi(\tau,k)=\mathcal{K}(\varphi(\tau),\varphi(\tau),\varphi(\tau))(k),\\\
&\varphi(0,k)=\varphi_{\mathrm{in}}(k),\end{split}\right.$ (WKE)

where $\varphi_{\mathrm{in}}$ is as above and $k\in\mathbb{R}^{d}$, and the
nonlinearity $\mathcal{K}$ (i.e. cubic collision operator) is given by

$\displaystyle\mathcal{K}(\varphi_{1},\varphi_{2},\varphi_{3})(k)$
$\displaystyle=\int_{(\mathbb{R}^{d})^{3}}\big{\\{}\varphi_{1}(k_{1})\varphi_{2}(k_{2})\varphi_{3}(k_{3})-\varphi_{1}(k)\varphi_{2}(k_{2})\varphi_{3}(k_{3})$
$\displaystyle\qquad\qquad\qquad\quad+\varphi_{1}(k_{1})\varphi_{2}(k)\varphi_{3}(k_{3})-\varphi_{1}(k_{1})\varphi_{2}(k_{2})\varphi_{3}(k)\big{\\}}$
$\displaystyle\qquad\qquad\times\bm{\delta}(k_{1}-k_{2}+k_{3}-k)\bm{\delta}(|k_{1}|^{2}-|k_{2}|^{2}+|k_{3}|^{2}-|k|^{2})\,\mathrm{d}k_{1}\mathrm{d}k_{2}\mathrm{d}k_{3}.$
(COL)

Here and below $\bm{\delta}$ denotes the Dirac delta, and we define

$|k|^{2}:=\langle k,k\rangle,\quad\langle
k,\ell\rangle:=k^{1}\ell^{1}+\cdots+k^{d}\ell^{d},$

where $k=(k^{1},\cdots,k^{d})$ and $\ell=(\ell^{1},\cdots,\ell^{d})$ are
$\mathbb{Z}_{L}^{d}$ or $\mathbb{R}^{d}$ vectors.

It is proved in [31] that (WKE) is locally well-posed with Schwartz data. In
general, global well-posedness is not true; there are known examples of
finite-time blowup [26, 27, 9]. In fact such blowup, in the form of $\delta$
singularity, is expected under mild conditions on initial data, and is
connected to the formation of condensates. See Section 1.5.3 for more related
discussions.

####  1.1.2. The main result

Our main result is as follows.

######  Theorem 1.1.

Fix $d\geq 3$ and $\gamma\in(0,1)$ (or assume $\gamma=1$ but replace
$\mathbb{T}_{L}^{d}$ by a rectangular torus and assume a genericity condition
on its aspect ratios), and fix Schwartz initial data
$\varphi_{\mathrm{in}}\geq 0$. We also fix
$\tau_{*}\in(0,\tau_{\mathrm{max}})$, where $\tau_{\max}$ is the maximal time
of existence for (WKE) (which may be finite or $\infty$), see Proposition 3.2
for the precise definition. Consider the equation (NLS) with random initial
data (DAT), and assume $\alpha=L^{-\gamma}$ so that
$T_{\mathrm{kin}}=\frac{1}{2}L^{2\gamma}$.

Then, for sufficiently large $L$, (NLS) has a smooth solution $u(t,x)$ up to
time

$T=\tau_{*}\cdot\frac{L^{2\gamma}}{2}=\tau_{*}\cdot T_{\mathrm{kin}},$

with probability $\geq 1-e^{-(\log L)^{40d}}$. Moreover we have

$\sup_{t\in[0,T]}\sup_{k\in\mathbb{Z}_{L}^{d}}\left|\mathbb{E}\,|\widehat{u}(t,k)|^{2}-\varphi\bigg{(}\frac{t}{T_{\mathrm{kin}}},k\bigg{)}\right|\leq
L^{-\theta},$ (1.2)

where $\theta>0$ depends only on $(d,\gamma)$ (see (3.19) in Section 3.3.2),
$\widehat{u}$ is as in (1.1), and $\varphi(\tau,k)$ is the solution to (WKE).
In (1.2) and below we understand that the expectation $\mathbb{E}$ is taken
assuming (NLS) has a smooth solution on $[0,T]$, which is an event with
overwhelming probability.

######  Remark 1.2.

The choice of the equation (NLS) is not essential, and ideas of the proof work
for any semilinear dispersive equation modulo technicalities222For equations
with derivative nonlinearity or quasilinear equations (such as water wave), it
is physically natural to add vanishing dissipations to the system, or to put
ultraviolet cutoffs at very high frequencies that converge to the identity in
the kinetic limit. With such considerations, our method of proof also work for
these equations.. The reason we choose to work with (NLS) is that it can be
viewed as the universal Hamiltonian equation, in the sense that any other
Hamiltonian equation reduces to (NLS) in a certain limit regime of solutions
(see [63]). Moreover, the kinetic theory for (NLS) is closely related to both
classical and quantum Boltzmann equations. In fact, the integration domain in
the collision kernel in (COL) is the same as that in the Boltzmann equation
(see Section 1.2 below for further discussions). All this makes (NLS) the
perfect choice as the underlying system in our studies.

######  Remark 1.3.

A major significance of Theorem 1.1 is that it allows all the rich long-time
dynamics of (WKE), see Section 1.5.3, to manifest via (NLS). In the case when
(WKE) has finite time blowup, Theorem 1.1 is clearly optimal (unless we want
to extend beyond the blowup time). When (WKE) has global solution, we can
allow the $\tau_{*}$ in Theorem 1.1 to grow with $L$, and an interesting
question is to determine the optimal dependence of $\tau_{*}$ on $L$. In
general this depends on the long-time behavior of the solution to (WKE); for
uniformly bounded solutions, our proof yields $\tau_{*}\sim\log\log L$. This
may be improved to $\tau_{*}\sim\log L/\log\log L$ by refining the arguments
in the proof (we made no such effort in this paper).

######  Remark 1.4.

The role of the scaling law $\gamma$ has been vague in early physics
literature, but has recently been clarified by the authors in [15, 16, 17,
18]. For the equation (NLS) on an arbitrary torus (i.e. without imposing
genericity condition on its aspect ratios), the admissible range of scaling
laws is $\gamma\in(0,1)$, with the endpoint $\gamma=0$ also being admissible
in the _discrete_ setting (see [48]) and $\gamma=1$ also being admissible if
we impose genericity condition on the torus [15]. This full range is covered
in [18] and the current paper. Two specific scaling laws are of particular
importance: $\gamma=1$ ($T_{\mathrm{kin}}\sim L^{2}$) which is linked to the
notable Gibbs measure invariance problem in dimension $d=3$, and $\gamma=1/2$
($T_{\mathrm{kin}}\sim L$), which corresponds to the Boltzmann-Grad scaling in
the particle case, and is a natural scaling law in the inhomogeneous setting.
See [17, 18] for more discussions.

######  Remark 1.5.

As in the earlier papers [15, 16, 18], we can also obtain the accompanying
results to Theorem 1.1 pertaining to higher order statistics, using the same
arguments in [16]. These include propagation of chaos (which in fact directly
follows from the proof of Theorem 1.1), probability density evolution in the
case of non-Gaussian initial data, and derivation of the wave kinetic
hierarchy; see [16] for more details.

######  Remark 1.6.

Unlike in [15, 16, 18], here the Schwartz assumption in Theorem 1.1 is
necessary (at least the level of regularity and decay required for
$\varphi_{\mathrm{in}}$ should depend on $\tau_{*}$), because in our inductive
argument (Section 3.5), the norms involved in the bounds get successively
weaker with each iteration.

###  1.2. Background and history

The work of Boltzmann [7] in the 1870s marked the formal beginning of _kinetic
theory_. In general, it studies the evolution of macroscopic333In some of the
literature, a distinction is made between the _mesoscopic_ observables studied
in kinetic theory (such as the density function), and the macroscopic ones
obtained by taking averages of the mescoscopic quantitites (such as fluid
velocity). observables in the kinetic limit, where certain statistical
averages of a given microscopic system (e.g. $N$ colliding particles) are
studied in the limit of infinite system size ($N\to\infty$). In the classical
case of colliding particles, the microscopic dynamics is Newtonian and the
corresponding macroscopic limit is given by the _Boltzmann equation_.

The parallel theory for waves, under the name of _wave kinetic theory_ or
_wave turbulence theory_ , was initiated in the work of Peierls [53] in the
1920s. Here particles are replaced by wave modes (or wave packets), collisions
are replaced by nonlinear wave interactions such as (NLS), and the Boltzmann
equation is replaced by the wave kinetic equation. With advances in the last
century, the wave kinetic theory has now become a vast subject, which provides
systematic treatments of wave interactions and has major scientific
applications across various disciplines [41, 42, 66, 2, 65, 68, 3, 13, 29, 67,
57, 44], see Nazarenko [50] for a relatively modern survey. In particular, we
mention the influential papers of Hasselmann [41, 42] and Zakharov [66]: the
former initiates the study of kinetic theory for water waves, leading to
highly successful applications to oceanography. The latter one discovers the
_Zakharov spectrum_ in analogy to the Kolmogorov spectrum in hydrodynamic
turbulence, which gives the theory its common name of _wave turbulence_.

Another aspect of wave kinetic theory appears when we replace classical
particles in the Boltzmann picture with quantum particles (Bosons or
Fermions). This leads to the quantum Boltzmann or _Boltzmann-Nordheim
equation_ , see the papers of Nordheim [51] and Uehling-Uhlenbeck [64] in the
1920–30s. The quantum Boltzmann equation is closely related to both the
classical Boltzmann equation and the wave kinetic equation (WKE) associated
with (NLS); in fact its collision operator is an exact linear combination of
the other two, and all three collision operators are given by integrations in
the same domain. On another hand, one can also consider classical or quantum
particle systems where the nonlinear interactions is rescaled by the size of
the system, in what is called the _mean field limit_. This leads to Vlasov-
type equations in the classical setting, and nonlinear Schrödinger equations
in the quantum setting. Such mean-field limits are relatively better
understood, and we shall not elaborate on them in our discussion here (cf.
[60, 24, 46, 56, 43, 37, 35] for a sample of results and some review
articles).

Mathematically speaking, the main challenge is to rigorously derive the
kinetic equations, as well as subsequent diffusion and other asymptotic
limits, from the underlying microscopic systems. For classical particles, this
was explicitly stated in _Hilbert’s Sixth Problem_ back in 1900 [38]. The
state of art in this particle setting has been the rigorous derivation of the
Boltzmann equation for _short time_ , first due to Lanford [47], King [45],
and more recently completed an improved in Gallagher-Saint-Raymond-Texier [30]
(see also [54, 4, 5, 6] for some recent developments). The next step, which
would be the justification of Boltzmann equation for arbitrarily long time,
has been a major open problem in this subject since the 1975 paper of Lanford
[47] 444More can be said when the microscopic particle system is replaced by a
random collision model as in [52]..

On the wave turbulence side, the mathematical progress started much slower,
but is quickly catching up in recent years. Earlier results such as Spohn [59]
and Erdös-Yau [25] focused on linear problems where the Feynman diagram
expansion has a simpler structure. These were then extended beyond the kinetic
time scale in the breakthrough of Erdös-Salmhofer-Yau [23].

The next major advancement was Lukkarinen-Spohn [48]. Though only involving
equilibrium settings, it was the first one to treat the full nonlinear
problem, and inspired many of the subsequent studies. In nonlinear, off-
equilibrium settings, partial results were obtained by several different
groups including the authors [8, 14, 11, 12], which covered successively
better time scales but still fell short of the kinetic time
$T_{\mathrm{kin}}$.

The state of the art prior to this work has been the authors’ recent papers
[15, 16, 18]. These provide the first rigorous derivation of (WKE) from (NLS)
that reaches (a small multiple of) $T_{\mathrm{kin}}$, thus matching the state
of the art in the particle case [30]. We also mention a few results concerning
the inhomogeneous setting and stochastic variants etc., see [28, 21, 62, 36,
40, 22, 1, 49], and concerning the solution theory to (WKE), see [26, 27, 9,
31, 58, 10]. Finally, the derivation of the Boltzmann-Nordheim equation, even
for short time, is still completely open.

We conclude that, in all the _nonlinear_ settings discussed above, the results
have been limited to short time, even for the equilibrium result in [48]. In
this context, the current paper provides the first rigorous derivation of any
collisional nonlinear kinetic equation (particle or wave) that goes beyond the
short time perturbative setting.

###  1.3. The difficulty

As discussed in Section 1.2, in both particle and wave settings, the
justification of the kinetic approximation has been limited to short time, and
extending this to arbitrarily long time has been a major unsolved problem. The
only exception is Erdös-Salmhofer-Yau [23] (extending Erdös-Yau [25]), which
concerns a _linear_ equation.

The fundamental difficulty involved in these long-time problems is the
divergence of Taylor series, which we explain below. In general, there are two
approaches to the short-time justification of the kinetic approximation: the
_hierarchical approach_ [47, 45, 30, 23, 48], which relies on the infinite
hierarchy system satisfied by the $n$-th order statistics (i.e. $n$-th moments
or $n$-particle distributions), and the _dynamical approach_ [8, 14, 11, 12,
15, 16, 18], which relies on approximating the dynamics of the microscopic
system.

In the dynamical approach, the solution to the microscopic system (such as
NLS)) is approximated by high order Duhamel expansions; this is essentially a
Taylor series in time, which will have to converge if we want this approach to
be valid. In the hierarchical approach, a key challenge is to obtain
exponential a priori bounds for high order target quantities (i.e. $n$-th
order statistics for very large $n$), which also corresponds to the absolute
convergence of a certain Taylor-type series expansion. Thus in either case, as
long as the approach is based on the structure of time $0$ data, the result
cannot go beyond the radius of convergence of the Taylor series at time $0$.

For _linear_ equations such as [23] this is not a problem, because under
suitable assumptions the above radius of convergence is _always_ $\infty$.
However, even for the simplest nonlinear equation, it is very likely that a
regular solution exists on a longer time interval, but the radius of
convergence of the Taylor expansion at time $0$ is much smaller. In such
cases, it is not possible to go beyond this radius of convergence, by simply
setting up an argument “based on time $0$”. On the other hand, if we want to
exploit any “base time” $t_{*}$ greater than 0, then we are faced with the
difficulty that the data at time $t_{*}$ does not have independent states or
modes, and the behavior of the target quantities at time $t_{*}$ relies on the
full dynamics on $[0,t_{*}]$.

###  1.4. The strategy

Our main theorem, Theorem 1.1, resolves the difficulty in Section 1.3 and
completes the long-time justification of the kinetic limit in the wave
turbulence setting. This relies on the following strategy, which is summarized
here, and expanded upon in Section 2 below.

Idea 1: Shifting base times. We follow the dynamical approach. As discussed in
Section 1.3, to go beyond the radius of convergence, we need to shift the base
time, and perform expansions at later times $t>0$. In practice, for fixed
$\tau_{*}$ as in Theorem 1.1, we may choose $\delta$ small enough depending on
$\tau_{*}$, such that $\tau_{*}=\mathfrak{D}\delta$ with
$\mathfrak{D}\in\mathbb{Z}$, and the solution $\varphi(\tau,k)$ has a
convergent Taylor expansion on $[p\delta,(p+1)\delta]$ with base point
$p\delta$, for each $0\leq p<\mathfrak{D}$. This formally allows us to expand
$u(\tau\cdot T_{\mathrm{kin}})$ as a Taylor series of $u(p\delta\cdot
T_{\mathrm{kin}})$ for $\tau\in[p\delta,(p+1)\delta]$, so _if $u(p\delta\cdot
T_{\mathrm{kin}})$ were to have independent Fourier coefficients as
$u(0)=u_{\mathrm{in}}$ does in (DAT)_, then repeating the same arguments as in
[15, 16, 18] would immediately yield (1.2) for $t\in[p\delta,(p+1)\delta]$. Of
course, such independence is not true due to the nonlinear evolution (NLS) on
$[0,p\delta\cdot T_{\mathrm{kin}}]$; but this is exactly the first central
component of our proof, namely to examine the sense in which the Fourier
coefficients of $u(p\delta\cdot T_{\mathrm{kin}})$ can be viewed as nearly
independent. This condition has to be strong enough to imply the validity of
kinetic approximation, but also has to be verified by the actual time
evolution. Moreover, this condition is to be proved _by induction in $p$_,
which means that it has to get successively weaker and involve successively
lower order quantities, as $p$ increases.

Idea 2: Quantifying “near independence”. Now the question becomes how to
measure the dependence of Fourier coefficients of $u(p\delta\cdot
T_{\mathrm{kin}})$. One possibility, suggested by the proof of propagation of
chaos in [16], is to express this in terms of mixed moments, or more precisely
_cumulants_ , which quantify how far the behavior of the mixed higher order
moments is from independence and Gaussianity.

Recall the definition of cumulants $\mathbb{K}(X_{1},\cdots,X_{r})$ (see [48],
or Definition 3.1 below). Now consider

$\mathbb{K}\big{(}\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k_{1})^{\zeta_{1}}\,,\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k_{2})^{\zeta_{2}}\,,\cdots,\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k_{r})^{\zeta_{r}}\big{)},$ (1.3)

where $k_{j}\in\mathbb{Z}_{L}^{d}$ and $\zeta_{j}\in\\{\pm\\}$ (indicating
possible conjugates, see Section 3.3.3 below). By gauge and space translation
invariance, it is easy to see that (1.3) vanishes unless
$\sum_{j}\zeta_{j}=\sum_{j}\zeta_{j}k_{j}=0$ (so $r$ must be even); moreover
(1.3) equals $\mathbb{E}|\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k)|^{2}\approx\varphi(p\delta,k)$ if $r=2$ and
$k_{1}=k_{2}=k$. As such, we know that the independence (and Gaussianity) of
the Fourier coefficients of $u(p\delta\cdot T_{\mathrm{kin}})$ exactly
corresponds to the vanishing of the cumulants (1.3) for $r\geq 4$.

Now, for near independence, it is natural to expect that such vanishing should
be replaced by suitable smallness or decay conditions for (1.3). Indeed, it
follows from the proof in [16] that

$|(\ref{introcm})|\leq L^{-cr},\quad\forall r\geq 4$ (1.4)

for some small value $c>0$. Unfortunately, this is too weak as we shall see in
an example below. On the other hand, in the Gibbs measure case [48], (1.4) is
replaced by the much stronger _physical space_ $L^{1}$ bound for cumulants;
however, this is too strong and a calculation shows that such bounds cannot
hold away from equilibrium (see Appendix B).

In fact, after some trial and error, we realize that having only upper bonds
on (1.3) is not enough to run the inductive argument. This is demonstrated by
an example, which we leave to Section 2.1 as it requires some preliminaries.
Basically, for $p=1$, we can find a term $\mathbb{K}$ in (1.3) with $r=4$,
such that when we expand $\widehat{u}(2\delta\cdot T_{\mathrm{kin}})$ into
Taylor series consisting of multilinear expressions of
$\widehat{u}(\delta\cdot T_{\mathrm{kin}})$, the contribution of $\mathbb{K}$
to $\mathbb{E}|\widehat{u}(2\delta\cdot T_{\mathrm{kin}},k)|^{2}$ is a term
that _does not appear in_ the leading term approximation $\varphi(2\delta,k)$,
but _contains the same power of_ $L$ as the approximation. Of course, this
anomaly still vanishes in the limit, but only because of the extra
cancellations caused by the _exact structure_ of $\mathbb{K}$. Such
structures, as will be shown in Section 2 below, reflects a certain _Markovian
property_ of solutions to the random data problem for (NLS).

Idea 3: Setting up the ansatz. We now need to set up the precise ansatz for
the cumulants (1.3), which involves not only upper bounds, but also exact
structures. Indeed, this ansatz will contain _a separate expansion_ for each
term (1.3) up to some high order, plus a remainder that is sufficiently small.

Schematically, we may choose $R_{p}\gg R_{p+1}$, and consider the ansatz

$\mathbb{K}\big{(}\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k_{1})^{\zeta_{1}}\,,\cdots,\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k_{r})^{\zeta_{r}}\big{)}=\sum(\mathrm{terms\ at\ step\
}p)+(\mathrm{small\ remainder})$ (1.5)

for all $r\leq R_{p}$; the inductive argument amounts to proving that if (1.5)
is true for $p$, then it is true for $p+1$. Of course, this is based on the
fact that $\widehat{u}((p+1)\delta\cdot T_{\mathrm{kin}})$ can be expanded
into Taylor series consisting of multilinear expressions of
$\widehat{u}(p\delta\cdot T_{\mathrm{kin}})$. As in all our previous works,
this Taylor series is organized into ternary trees; after using the technical
Lemma A.1, we then obtain a “pre-Feynman diagram” expansion of the cumulants
(1.3) for $p+1$ in terms of those for $p$, which is organized into multiple
ternary trees.

The exact ansatz (1.5) is then determined by inductively iterating the above
pre-expansion. By looking at the first few $p$’s, we may guess that (1.5) can
be organized into _gardens_ (i.e. multiple trees with their leaves paired, see
[16] or Definition 3.4 below). These gardens $\mathcal{G}$ at level $p+1$
(which we denote by $\mathcal{G}_{p+1}$) can be constructed by induction as
follows. Consider the multiple ternary trees occurring in the pre-expansion
above; upon using Lemma A.1, their leaves are divided into two-element pairs
(corresponding to the two-point cumulant $\mathbb{E}|\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k_{\mathfrak{l}})|^{2}$) and four-or-more-element subsets
(corresponding to higher order cumulants (1.3) for $p$ with $r\geq 4$). We
then construct the garden $\mathcal{G}_{p+1}$ by keeping all the leaf pairs,
and replacing each four-or-more-element subset by a garden of form
$\mathcal{G}_{p}$.

This strategy can be described as a “partial time series expansion”, i.e. we
stop the time series expansion at time $p\delta$ if we encounter a two-point
cumulant, and continue with the expansion if we encounter higher order
cumulants. We replace the two-point cumulant
$\mathbb{E}|\widehat{u}(p\delta\cdot T_{\mathrm{kin}},k_{\mathfrak{l}})|^{2}$
by its approximation $\varphi(p\delta,k_{\mathfrak{l}})$, and replace the
higher order cumulants by the induction hypothesis in (1.5). This partial time
series expansion has critical importance in our proof, as it allows us to make
use of the uniform bound of $\varphi$ _outside the radius of convergence at
$t=0$_, and avoid falling into the trap of divergent Taylor series.

Eventually, these iteration steps lead to completely new combinatorial
structures, which we call layered gardens (Definition 3.6) and canonical
layered gardens (Definition 3.7). We need to work out all the properties of
these objects from scratch, including the combinatorial ones (Propositions
4.8–4.10, 7.19, 8.2–8.3) and the analytic ones (Propositions 5.1, 6.1–6.3,
9.5, 10.4–10.5). These are the most central parts of this paper. Once these
are done, we will be in a position to apply some of the arguments from [15,
16, 18] _as a black box_ , to conclude the inductive proof. For a detailed
discussion of the example, the motivation and the ansatz, see Section 2 below.

###  1.5. Future horizons

Theorem 1.1 and its proof opens a door to many problems that were previously
out of reach. In this section we only mention the three most important avenues
that should be now open for investigation.

####  1.5.1. Extension of Lanford’s theorem

With the analogy drawn between particle and wave kinetic theory, it is natural
to expect that the particle counterpart of Theorem 1.1, i.e. the long-time
extension of Lanford’s theorem [47, 45, 30], would also be true. The authors
are investigating this in an upcoming paper [19] jointly with Xiao Ma.

####  1.5.2. Fluctuations and large deviations

The cumulant estimates in this paper should be sufficient to obtain the
fluctuation and large deviation dynamics around the wave kinetic limit given
by (WKE), over the same long time interval as in Theorem 1.1. Such fluctuation
and large deviation dynamics were described from a physics point of view in
[33]. The parallel results in the particle setting were proved recently in [4]
for short times.

####  1.5.3. Diffusion and other asymptotic limits

Once the kinetic approximation is justified for arbitrarily long time, the
next step in the Hilbert’s Sixth Problem, in both particle and wave settings,
is then to reach the time scale for diffusion and other asymptotic limits.

More precisely, in the case of Boltzmann equation, it is known that certain
rescaled limits of solutions converge to solutions to fluid equations, such as
the Euler and Navier-Stokes equations, see [34, 55, 32] and references
therein. Deriving those fluid equations directly from Newtonian dynamics would
then require extending Lanford’s theorem to the adequate time scale of this
limit.

In the case of _inhomogeneous_ wave kinetic equation or Boltzmann-Nordheim
equation555We remark that the derivation of these inhomogeneous equations has
not been complete, at least in the setting without noise (see [1, 40, 39] for
some related progress); however we expect this to follow from similar
arguments as in [15, 16, 18] and the current paper, modulo technical
differences., i.e. when $\varphi=\varphi(\tau,x,k)$ in (WKE) and
$\partial_{\tau}\varphi$ is replaced by
$(\partial_{\tau}+k\cdot\nabla_{x})\varphi$, it is not clear whether a
diffusion limit similar to Boltzmann exists. What is the possible asymptotic
behavior of the kinetic equation, and what this may imply about the behavior
of (NLS) _beyond the kinetic time scale_ , is a major open avenue of
investigation in this subject.

Finally, for the _homogeneous_ wave kinetic equation or the Boltzmann-Nordheim
equation, there are known finite time blowups where
$\tau_{\mathrm{max}}<\infty$ with formation of $\delta$ singularity (see [26,
27, 9]). In fact this is expected to be fairly general, and corresponds to the
formation of condensates. It may be possible to continue (WKE) weakly after
the blowup time, but the sense in which the approximation (1.2) holds probably
needs to be modified. How to make this rigorous is another outstanding open
problem.

There also exist (formal) global power-law solutions to (WKE) known as the
Kolmogorov-Zakharov cascade spectra, which were recently investigated in [10].
These solutions are constant-flux cascade solutions for (WKE), and their
rigorous understanding is yet another fascinating question. Once this cascade
behavior is understood at the level of (WKE), Theorem 1.1 will immediately
allow to prove energy cascade at the level of (NLS), i.e. rigorously establish
the generic cascade phenomenon for nonlinear Schrödinger equations. This links
to the extensively studied, yet still open, question of growth of Sobolev
norms for (NLS) and similar nonlinear dispersive equations.

### Acknowledgements

The authors are partly supported by a Simons Collaboration Grant on Wave
Turbulence. The first author is supported in part by NSF grant DMS-2246908 and
Sloan Fellowship. The second author is supported in part by NSF grant
DMS-1936640.

##  2\. Description of main ideas

In this section we detail out some of the key ideas leading to the main ansatz
(Proposition 3.9), which was briefly stated in Section 1.4. Throughout this
section we will use the (now standard) concepts of trees, couples, gardens and
decorations etc., see Figure 1; the reader may consult the authors’ earlier
papers [15, 16, 18] for the definitions of these terms, or see Definitions
3.3–3.5 below.

Figure 1. From left to right: a tree, a couple and a garden of $4$ trees, with
corresponding decorations (leaves of same color indicate pairings). See
Definitions 3.3–3.5 below or [15, 16, 18] for precise definitions.

###  2.1. An example

We start by discussing the example stated in Section 1.4, Idea 2. Recall that
the Fourier coefficients $\widehat{u}(0,k)$ are independent for different $k$,
but $\widehat{u}(\delta\cdot T_{\mathrm{kin}},k)$ are no longer independent
due to the nonlinear evolution. The first deviation from independence is
characterized by the order $4$ cumulant, say

$\mathbb{K}\big{(}\overline{\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{1})},\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{2}),\overline{\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{3})},\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{4})\big{)}.$ (2.1)

Recall that $\widehat{u}(\delta\cdot T_{\mathrm{kin}})$ can be expanded into a
Taylor series consisting of multilinear expressions of $\widehat{u}(0)$, where
these terms are organized using ternary trees (see Definition 3.3, and (3.33)
and (3.35) below). If all four factors in (2.1) are replaced by the _linear_
evolution, then the independence at time $0$ will carry over and the resulting
cumulant will be $0$; thus the first nonzero contribution to (2.1) is given by
the case when three of the four factors are replaced by the linear evolution,
and the remaining one is replaced by the cubic term (i.e. the first iterate,
which corresponds to the ternary tree with only one branching node).

We know the tree expansion is calculated by iterating the nonlinearity in
(NLS), for example

$\widehat{u}(t,k_{1})\sim e^{2\pi
i|k_{1}|^{2}t}\bigg{[}\widehat{u}(0,k_{1})+iL^{-d}\alpha\sum_{k_{2}-k_{3}+k_{4}=k_{1}}\int_{0}^{t}e^{2\pi
i\Omega
s}\,\mathrm{d}s\cdot\widehat{u}(0,k_{2})\overline{\widehat{u}(0,k_{3})}\widehat{u}(0,k_{4})\bigg{]}$
(2.2)

(we only show the first two terms in the expansion), where
$\Omega:=|k_{2}|^{2}-|k_{3}|^{2}+|k_{4}|^{2}-|k_{1}|^{2}$. Now, if we replace
the first factor in (2.1) by the cubic term, and apply Lemma A.1, we would
obtain (part of) the first nonzero contribution to (2.1), which has the form

$\mathbb{K}_{0}(k_{1},k_{2},k_{3},k_{4}):=e^{2\pi i\Omega\cdot\delta
T_{\mathrm{kin}}}\cdot
L^{-d-\gamma}\cdot\mathbf{1}_{k_{1}-k_{2}+k_{3}-k_{4}=0}\cdot\prod_{j=2}^{4}\varphi_{\mathrm{in}}(k_{j})\cdot\int_{0}^{\delta\cdot
T_{\mathrm{kin}}}e^{-2\pi i\Omega s}\,\mathrm{d}s.$ (2.3)

Note that this term (as well as the other terms in the full cumulant expansion
(2.1)) is bounded pointwise by $L^{-(d-\gamma)}$; however, knowing _only_ this
upper bound will not be sufficient, which is what we will demonstrate through
this example.

To see this, consider the expansion similar to (2.2) but with starting time
$\delta\cdot T_{\mathrm{kin}}$ rather than 0. Now we calculate the quantity
$\mathbb{E}|\widehat{u}(2\delta\cdot T_{\mathrm{kin}},k)|^{2}$, in which we
replace one of the two factors by the linear evolution of
$\widehat{u}(\delta\cdot T_{\mathrm{kin}})$, and the other factor by the cubic
term involving $\widehat{u}(\delta\cdot T_{\mathrm{kin}})$. This then leads to
the contribution

$\mathbb{Y}:=L^{-d-\gamma}\sum_{k_{1}-k_{2}+k_{3}=k}e^{-2\pi
i\Omega\cdot\delta T_{\mathrm{kin}}}\cdot\int_{\delta\cdot
T_{\mathrm{kin}}}^{2\delta\cdot T_{\mathrm{kin}}}e^{2\pi i\Omega
t}\,\mathrm{d}t\cdot\mathbb{K}_{0}(k,k_{1},k_{2},k_{3}),$ (2.4)

where $\Omega$ is defined for $(k,k_{1},k_{2},k_{3})$. In this sum we may
restrict to the region $|\Omega|\leq L^{-2\gamma}$, so the $\mathrm{d}t$
integral has saturated size $\sim T_{\mathrm{kin}}$; a dimension counting
argument then yields that the number of choices of $(k_{1},k_{2},k_{3})$ for
fixed $k$ is at most $L^{2(d-\gamma)}$ (see Lemma A.9). Now, if we only have
the upper bound $L^{-(d-\gamma)}$ of $\mathbb{K}_{0}$, then restricting the
sum in (2.4) to the region $|\Omega|\leq L^{-2\gamma}$ yields a sum whose
upper bound is $L^{-d-\gamma}\cdot T_{\mathrm{kin}}\cdot L^{2(d-\gamma)}\cdot
L^{-(d-\gamma)}\sim 1$, which has _the same size_ as the leading term; however
it is clearly _not_ a part of the leading term approximation to
$\mathbb{E}|\widehat{u}(2\delta\cdot T_{\mathrm{kin}},k)|^{2}$.

The conclusion is that, it is impossible to exclude the “fake leading term”
$\mathbb{Y}$ and obtain the correct asymptotics, if we only use the upper
bound for $\mathbb{K}_{0}$ defined in (2.3). To eliminate $\mathbb{Y}$ in the
limit, we need to use the exact structure (2.3), which (after a time
rescaling) leads to the expression

$\mathbb{Y}=L^{-2(d-\gamma)}\sum_{k_{1}-k_{2}+k_{3}=k}\prod_{j=1}^{3}\varphi_{\mathrm{in}}(k_{j})\cdot\int_{0<s<\delta<t<2\delta}e^{\pi
i\Omega\cdot L^{2\gamma}(t-s)}\,\mathrm{d}t\mathrm{d}s.$ (2.5)

Now $\mathbb{Y}$ disappears in the limit, because for fixed $(t,s)$, the
asymptotics of the summation in $(k_{1},k_{2},k_{3})$ contains a factor
$\bm{\delta}(t-s)$, whose integral in $(t,s)$ then vanishes due to the
restriction $s<\delta<t$.

The above cancellation mechanism is similar to the one exploited in [18]
involving vines (II), which is also used in the current paper. On the other
hand, the vanishing of $\mathbb{Y}$ is linked to the fact that the intervals
$[0,\delta]$ and $[\delta,2\delta]$ have no common interior; this draws
interesting analogy with the properties of Markov processes (for example, the
Brownian motion whose increments on disjoint intervals are independent and
have vanishing correlation). In fact, what the above analysis demonstrates is
exactly a kind of _Markovian property_ ; namely, the asymptotic two-point
correlations at time $2\delta\cdot T_{\mathrm{kin}}$ depend only on the
asymptotic two-point correlations at time $\delta\cdot T_{\mathrm{kin}}$, but
_not_ on the nonlinear evolution history from earlier times which manifests as
the higher order cumulants $\mathbb{K}_{0}$.

###  2.2. Building up gardens: A naive attempt

Recall the notations introduced in Section 1.4 and our choice of $\delta$ so
that the interval $[0,\tau_{*}]$ is split into $\mathfrak{D}$ subintervals of
length $\delta$. As explained in Section 1.4, a central step in estimating
$\mathbb{E}|\widehat{u}(p\delta\cdot T_{\mathrm{kin}},k)|^{2}$, and hence in
the proof of Theorem 1.1, is to analyze and control the exact structure of the
cumulants of form (1.3) (these cumulants are related to the corresponding
moments as in Definition 3.1 below). The example in Section 2.1 already
illustrates the process of calculating moments and cumulants at time
$(p+1)\delta\cdot T_{\mathrm{kin}}$, using those at time $p\delta\cdot
T_{\mathrm{kin}}$ or earlier. Generalizing this procedure naturally leads to
the construction of _layered gardens and couples_ (i.e. each node is in a
unique layer indicated by nonnegative integers), as briefed in Section 1.4,
Idea 3. To motivate, in this section we will first describe a naive but
incorrect attempt, and leave the real construction, which is only one step
away, to Section 2.3.

Recall the $R_{p}$ described in Section 1.4; let $N_{p}$ be such that
$N_{p}\gg R_{p}\gg N_{p+1}$. For each $p<\mathfrak{D}$, we can always perform
an order $N_{p+1}$ Duhamel expansion of the solution at time $(p+1)\delta\cdot
T_{\mathrm{kin}}$, which expresses it as a sum of iterates of the time
$p\delta\cdot T_{\mathrm{kin}}$ data plus a remainder term. The iterates are
multilinear expressions in the solution at time $p\delta\cdot
T_{\mathrm{kin}}$ and can be arranged as sums of ternary trees whose order
(i.e. number of branching nodes) equal the order of the iteration (cf.
equations (3.36), (3.33), and Definitions 3.3 and 3.5). Note that for $p=0$,
all the cumulants

$\mathbb{K}\big{(}\widehat{u}(0,k_{1})^{\zeta_{1}}\,,\widehat{u}(0,k_{2})^{\zeta_{2}}\,,\cdots,\widehat{u}(0,k_{r})^{\zeta_{r}}\big{)}$

at time $0$ and of order $r\geq 4$ should vanish due to exact independence and
Gaussianity in (DAT).

For $p=1$, by performing the above expansion, and applying Lemma A.1, we can
write the main (non-remainder) contribution to the cumulant

$\mathbb{K}\big{(}\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{1})^{\zeta_{1}}\,,\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{2})^{\zeta_{2}}\,,\cdots,\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{r})^{\zeta_{r}}\big{)}$ (2.6)

as a sum over _irreducible gardens_ $\mathcal{G}$ of width $r$. Here an
irreducible garden is a garden of width $r$ (i.e. a collection of $r$ trees
whose leaves are completely paired with each other, see Definition 3.4 below),
such that no proper subset of these $r$ trees have their leaves completely
paired. Note that this irreducibility property precisely corresponds to the
definition of the cumulant $\mathbb{K}$ in (2.6); instead, the corresponding
moment in (2.6) would consist of the sum over all (reducible and irreducible)
gardens. This gives the explicit structure of the cumulants at time
$\delta\cdot T_{\mathrm{kin}}$ as a sum over irreducible gardens
$\mathcal{G}$, which we denote by666We use the notion $\mathscr{N}$ because
the construction here is _not_ the correct ansatz, but a naive attempt; see
Section 2.3 for more discussions. $\mathcal{G}\in\mathscr{N}_{1}$. We shall
put all the nodes in this garden $\mathcal{G}\in\mathscr{N}_{1}$ into layer
$0$; then each term occurring in this summation is of form
$\mathcal{K}_{\mathcal{G}}$ which consists of a summation over all decorations
$(k_{\mathfrak{n}})$ and an integration over time variables
$(t_{\mathfrak{n}})$ defined by the layering (which are all $0$ for
$\mathcal{G}\in\mathscr{N}_{1}$). See (3.24) in Definition 3.8 below for the
exact expression $\mathcal{K}_{\mathcal{G}}$; note that all input functions
$F_{\mathfrak{L}_{\mathfrak{l}}}(k_{\mathfrak{l}})$ should be replaced by
$\varphi_{\mathrm{in}}(k_{\mathfrak{l}})=\varphi(0,k_{\mathfrak{l}})$.

Now let us move to $p=2$. Again we express the solution
$\widehat{u}(2\delta\cdot T_{\mathrm{kin}},k)$ as a Duhamel expansion, but now
_with initial time_ $\delta\cdot T_{\mathrm{kin}}$. Up to a remainder term
that we ignore for now, this writes $\widehat{u}(2\delta\cdot
T_{\mathrm{kin}},k)$ as a sum of terms associated with ternary trees whose
leaves are associated with $\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{\mathfrak{l}})$. As such, this allows to express the
cumulant

$\mathbb{K}\big{(}\widehat{u}(2\delta\cdot
T_{\mathrm{kin}},k_{1})^{\zeta_{1}},\cdots,\widehat{u}(2\delta\cdot
T_{\mathrm{kin}},k_{r})^{\zeta_{r}}\big{)}=\sum\mathbb{K}\big{(}(\mathcal{J}_{\mathcal{T}_{1}})_{k_{1}},\cdots,(\mathcal{J}_{\mathcal{T}_{r}})_{k_{r}}\big{)}$
(2.7)

where the sum is taken over all trees $\mathcal{T}_{j}$ of sign $\zeta_{j}$
and order at most $N_{2}$, and each $\mathcal{J}_{\mathcal{T}_{j}}$ is the
Duhamel iterate associated with the interaction history described by
$\mathcal{T}_{j}$, whose exact expression is as in (3.33) below. Then, the
cumulants in the sum in (2.7) can be written as linear combinations of

$\mathbb{K}\bigg{(}\prod_{\mathfrak{l}\in\mathcal{L}_{1}}\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{\mathfrak{l}})^{\sigma_{\mathfrak{l}}},\cdots,\prod_{\mathfrak{l}\in\mathcal{L}_{r}}\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{\mathfrak{l}})^{\sigma_{\mathfrak{l}}}\bigg{)},$ (2.8)

where $\mathcal{L}_{j}$ denotes the set of leaf nodes of the tree
$\mathcal{T}_{j}$ and $\sigma_{\mathfrak{l}}\in\\{\pm\\}$ is the sign of each
leaf $\mathfrak{l}$. By applying Lemma A.1 again, we an express (2.8) as a sum
of form

$\sum_{\mathcal{P}}\prod_{B\in\mathcal{P}}\mathbb{K}\big{(}(\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{\mathfrak{l}})^{\sigma_{\mathfrak{l}}})_{\mathfrak{l}\in
B}\big{)}.$ (2.9)

Here the summation in (2.9) is taken over all irreducible partitions
$\mathcal{P}$ of the set $\mathcal{L}_{1}\cup\cdots\mathcal{L}_{r}$ (i.e. no
nontrivial union of sets in $\mathcal{P}$ equals any nontrivial union of sets
$\mathcal{L}_{j}$), and the cumulant for each $B$ involves the random
variables $\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{\mathfrak{l}})^{\sigma_{\mathfrak{l}}}$ for
$\mathfrak{l}\in B$.

Now, we can use the formula deduced above for the cumulants (2.6) at time
$\delta\cdot T_{\mathrm{kin}}$, namely the expansion into terms associated
with $\mathcal{G}\in\mathscr{N}_{1}$, to replace each individual term in
(2.9), and subsequently obtain an expression for (2.7). Combinatorially, this
expression is associated with a new, bigger garden $\mathcal{G}_{2}$, which is
obtained as follows: start from the collection of trees
$(\mathcal{T}_{j}:1\leq j\leq r)$, then for each set $B\in\mathcal{P}$ of
leaves, we attach a garden $\mathcal{G}=\mathcal{G}(B)\in\mathscr{N}_{1}$ at
these leaves, so that these leaves become the roots of the trees in this
garden. Moreover, we put all _branching nodes_ of the trees $\mathcal{T}_{j}$
into layer $1$, and keep all nodes of each $\mathcal{G}(B)$ in layer $0$, to
make $\mathcal{G}_{2}$ a layered garden, and denote all such layered gardens
by $\mathscr{N}_{2}$. Analytically, by the definitions of
$\mathcal{J}_{\mathcal{T}}$ and $\mathcal{K}_{\mathcal{G}}$ (see (3.33) and
(3.24) below), it is easy to see that the expression associated with
$\mathcal{G}_{2}$ is precisely $\mathcal{K}_{\mathcal{G}_{2}}$ in (3.24)
below, where again the time integration is defined by the layering, and all
inputs $F_{\mathfrak{L}_{\mathfrak{l}}}(k_{\mathfrak{l}})$ should be replaced
by $\varphi(0,k_{\mathfrak{l}})$.

A simple instance of the above construction is already contained in the
example described in Section 2.1, where we have two trees
$(\mathcal{T}_{1},\mathcal{T}_{2})$ with one being trivial (only one leaf) and
the other having exactly one branch and three leaves, and attach the garden in
the right of Figure 1, to obtain the couple in the middle of Figure 1 with
suitable layering; note that the garden we attach has _width $4$_. Another
instance of this construction is given in Figure 2, where we have two trees
$(\mathcal{T}_{1},\mathcal{T}_{2})$ each with two branches, and each of the
five gardens in $\mathscr{N}_{1}$ we attach has _width $2$_ (i.e. is a couple)
with three being trivial and two being nontrivial.

Figure 2. An example of a layered couple as described in Section 2.2 which
appears in the expansion of $\widehat{u}(2\delta\cdot T_{\mathrm{kin}})$ (the
numbers [0] or [1] denote the layer of each node). This is a regular couple
(see Definition 4.1 below) but is _not_ canonical (see Section 2.3). In the
modified construction in Section 2.3, this will be replaced by the canonical
regular couple in Figure 3.

At this point, it is now clear how to continue this process inductively and
construct $\mathscr{N}_{p+1}$ from $\mathscr{N}_{p}$ etc., which ultimately
gives (up to acceptable remainder terms) an exact _structural_ description of
the cumulants at any time $p\delta\cdot T_{\mathrm{kin}}$, as a sum of terms
of form $\mathcal{K}_{\mathcal{G}}$ over all irreducible layered gardens
$\mathcal{G}\in\mathscr{N}_{p}$. This clearly accounts for the example in
Section 2.1; however this naive attempt still does not give the correct
ansatz, due to one fatal weakness. We discuss this weakness, as well as the
one simple step to fix it, in Section 2.3 below.

###  2.3. Canonical layering and the true ansatz

An indication why the construction in Section 2.2 does not work is that, by
this construction, all leaf nodes _must_ be in layer $0$ (in other words, all
the inputs in $\mathcal{K}_{\mathcal{G}}$ in (3.24) must be replaced by the
initial data $\varphi(0,k_{\mathfrak{l}})$), which is exactly what we want to
avoid in Section 1.3 (i.e. base all the analysis on time $0$ data).

To see an instance of this major issue, we take a closer look at the layered
couple $\mathcal{Q}$ in Figure 2. Note that it is a _regular couple_ (see
Definition 4.1 below), and does not incorporate the cancellation shown in the
example in Section 2.1 (i.e. it is _coherent_ in the terminology of Definition
4.7 below); thus there is no gain of powers of $L$ in the associated
$\mathcal{K}_{\mathcal{Q}}$ expression. Moreover, if the layer $1$ nodes in
$\mathcal{Q}$ are replaced by layer $p$, then the roots of the two sub-couples
with dashed lines can have _arbitrary_ layer between $0$ and $p$ while still
being consistent with the construction in Section 2.2. Therefore, the _number
of layerings_ of a given regular couple with order $2n$ can be as large as
$\mathfrak{D}^{n}$. Since the $\mathcal{K}_{\mathcal{Q}}$ quantity for each
individual regular couple $\mathcal{Q}$ gains only a factor of
$(C_{1}\delta)^{n}$ (where $C_{1}$ is a large constant depending on initial
data $\varphi_{\mathrm{in}}$, in the same way as in [15, 16, 18]), this will
lead to serious divergence issues as
$(C_{1}\mathfrak{D}\delta)^{n}=(C_{1}\tau_{*})^{n}$ grows in $n$ unless
$\tau_{*}$ is sufficiently small.

Of course, the above weakness comes from the fact that we are trying to
perform the Duhamel expansion all the way to time $0$ _in every case_. Indeed,
whenever we encounter the _two-point correlation_
$\mathbb{E}|\widehat{u}(p\delta\cdot T_{\mathrm{kin}},k_{\mathfrak{l}})|^{2}$
in the expression, we should simply replace it by its (expected) approximation
$\varphi(p\delta,k_{\mathfrak{l}})$, instead of further Duhamel-expanding it
by earlier time data and going all the way to time $0$. In this way we can
exploit the fact that $\varphi(p\delta,k_{\mathfrak{l}})$ is uniformly bounded
in $p$, despite that it cannot be expanded as a power series involving only
$\varphi_{\mathrm{in}}=\varphi(0,\cdot)$.

Now, recall the reduction and construction process described in Section 2.2.
By Lemma A.1, it is clear that, we would encounter the two-point correlation
$\mathbb{E}|\widehat{u}(p\delta\cdot T_{\mathrm{kin}},k_{\mathfrak{l}})|^{2}$
_only when_ one of the sets $B\in\mathcal{P}$ as in Section 2.2 has exactly
cardinality $2$. This leads to the following minor but crucial modification to
the construction process in Section 2.2, which is consistent with the brief
description in Section 1.4, Idea 3: namely, in reducing (2.9) (and same with
$\delta$ replaced by arbitrary $p\delta$), we further expand the cumulant
$\mathbb{K}\big{(}((\widehat{u}(\delta\cdot
T_{\mathrm{kin}},k_{\mathfrak{l}}))^{\sigma_{\mathfrak{l}}})_{\mathfrak{l}\in
B}\big{)}$ using earlier time ansatz _only when_ $|B|\geq 4$. When $|B|=2$, we
simply replace the cumulant, which is now just
$\mathbb{E}|\widehat{u}(\delta\cdot T_{\mathrm{kin}},k_{\mathfrak{l}})|^{2}$,
by $\varphi(\delta,k_{\mathfrak{l}})$ plus a decaying sub-leading term. Note
that this minor modification addresses both issued mentioned above: now we are
stopping expansion at time $p\delta\cdot T_{\mathrm{kin}}$ whenever we
encounter a two-point correlation at this time, and a combinatorial argument
allows us to control the number of coherent regular couples (see Section 2.3.1
below).

Combinatorially, the new construction leads to a new class of layered gardens
(and couples) $\mathcal{G}$, which we refer to as _canonical_ layered gardens
or $\mathcal{G}\in\mathscr{G}_{p}$, as follows: for $p=1$, $\mathscr{G}_{1}$
still denotes the set of all irreducible gardens with all nodes in layer $0$
(which is same as $\mathscr{N}_{1}$ in Section 2.2); for $p\geq 1$,
$\mathscr{G}_{p+1}$ is constructed from $\mathscr{G}_{p}$ in the same way as
$\mathscr{N}_{p+1}$ is constructed from $\mathscr{N}_{p}$ in Section 2.2,
except that we attach the garden $\mathcal{G}(B)\in\mathscr{G}_{p}$ _only
when_ $|B|\geq 4$, while for $|B|=2$ we simply _turn these two leaves into a
leaf pair and put both of them in layer $p$_.

For example, the example described in Section 2.1 is canonical, because the
only garden we attach has width $4$; on the other hand, the layered garden in
Figure 2 is not canonical, as the attached gardens all have width $2$. Indeed,
starting from two trees each with 2 branches as in Section 2.2, instead of
attaching the five width $2$ gardens (i.e. couples), the new construction
procedure requires to turn each two-leaf set $B$ into a leaf pair in layer
$1$, which results in the garden as illustrated in Figure 3, which is a
canonical layered garden in $\mathscr{G}_{2}$. See also Figures 4 and 5 for a
canonical garden in $\mathscr{G}_{3}$ and its construction process.

Figure 3. The canonical (regular) couple we get, if we start from the same
setup in Figure 2 in Section 2.2, but follow the new construction procedure in
Section 2.3. Here we do not attach any width $2$ garden (i.e. couple) but turn
any two-leaf subset $B$ into a leaf pair in layer $1$.

It can be shown, see Proposition 4.8, that a layered garden is canonical if
and only if for any two nodes $(\mathfrak{n},\mathfrak{n}^{\prime})$ such that
all their descendant leaves are completely paired, we always have
$\max(\mathfrak{L}_{\mathfrak{n}},\mathfrak{L}_{\mathfrak{n}^{\prime}})\geq\min(\mathfrak{L}_{\mathfrak{n}^{\mathrm{pr}}},\mathfrak{L}_{(\mathfrak{n}^{\prime})^{\mathrm{pr}}})$,
where $\mathfrak{L}_{\mathfrak{n}}$ is the layer of $\mathfrak{n}$ and
$\mathfrak{n}^{\mathrm{pr}}$ is the parent node of $\mathfrak{n}$ etc. This
provides another easy way to test if a given layered garden is canonical (for
example the garden in Figure 2 clearly violates this condition), and also
allows to bound the number of layerings satisfying the coherent condition in
Definition 4.7, see Proposition 4.10.

Analytically, with the new reduction process, and by similar arguments as in
Section 2.2, we see that the cumulants at time $p\delta\cdot T_{\mathrm{kin}}$
can be expressed as a sum of $\mathcal{K}_{\mathcal{G}}$ over all canonical
layered gardens $\mathcal{G}\in\mathscr{G}_{p}$, where
$\mathcal{K}_{\mathcal{G}}$ now is defined exactly as in (3.24), with time
integration defined by the layering, and the inputs given by777Compared to
Section 2.2, here the inputs are not replaced by the initial data
$\varphi(0,\cdot)$ because we may not be expanding all the way to time $0$.
$F_{\mathfrak{L}_{\mathfrak{l}}}(k_{\mathfrak{l}})$. Here
$\mathfrak{L}_{\mathfrak{l}}$ is the layer of $\mathfrak{l}$ and in practice
$F_{p}(k)$ equals $\varphi(p\delta,k)$ plus a decaying sub-leading term. For
an example (associated with the garden in Figures 4 and 5) of deducing the
expression $\mathcal{K}_{\mathcal{G}}$ for $\mathcal{G}\in\mathscr{G}_{3}$
from expressions $\mathcal{K}_{\mathcal{G}^{\prime}}$ for
$\mathcal{G}^{\prime}\in\mathscr{G}_{2}$, see Section 3.6.

####  2.3.1. Main components of the proof

With the key definition of canonical layered gardens introduced and the ansatz
for cumulants (1.3) fixed, we now describe the main components of the rest of
the proof. The proof requires to (i) estimate and analyze the expressions
$\mathcal{K}_{\mathcal{G}}$ for individual canonical layered gardens
$\mathcal{G}$ (or suitable combinations thereof), and (ii) control the number
of canonical layered gardens under suitable assumptions.

  * •

Non-regular couples: It is known since [15, 16, 18] that the _regular couples_
$\mathcal{G}=\mathcal{Q}$ (see Definition 4.1) provide the main contributions
to $\mathcal{K}_{\mathcal{G}}$. For non-regular couples $\mathcal{G}$, and
non-regular parts of arbitrary gardens and couples (apart from the special
structures of _vines_ discussed below), the corresponding
$\mathcal{K}_{\mathcal{G}}$ can be estimated by applying a particular counting
algorithm. This algorithm is a major component in the proofs of [15, 18], and
exactly the same algorithm and proof applies also in this paper. This is
summarized in the _rigidity theorem_ (Proposition 11.7), which we will use as
a black box.

  * •

Non-coherent regular couples: Now we restrict to regular couples. A key new
notion in this paper is that of _incoherency_ , see Definition 4.7 below. This
notion of incoherency relies on the layering structure, and basically
corresponds to the time disjointness property exhibited in the example in
Section 2.1; heuristically, such incoherency will occur in regular couples
each time we attach a garden of width at least $4$ to a set $B$ of leave as
described in Section 2.2 and above. Due to the Markovian property observed in
Section 2.1, we can prove that each occurrence of incoherency leads to a gain
of power $L^{-c}$ for some absolute constant $c$ (see Proposition 5.1), which
suffices for our proof.

  * •

Coherent regular couples: Turning to regular couples with no incoherency (i.e.
coherent), we can prove a structure theorem for them (see Propositions
4.8–4.9), which essentially reduces them to regular couples in a single layer
(as the one in Figure 3) which are already treated in [15, 18]. This allows us
to obtain the full asymptotics and cancellation for
$\mathcal{K}_{\mathcal{G}}$ associated with such couples
$\mathcal{G}=\mathcal{Q}$, see Propositions 6.1–6.3. In particular, the
leading contribution in $\mathbb{E}|\widehat{u}(p\delta\cdot
T_{\mathrm{kin}},k)|^{2}$ can be precisely calculated using these asymptotics,
which is then shown to be $\varphi(p\delta,k)$ by induction in time.

In addition, we can show that, for any coherent regular part of arbitrary
gardens and couples that has size $n$, the number of choices for this part,
including possible layerings, is bounded by $C_{0}^{n}$ where $C_{0}$ is a
constant _independent_ of $\delta$ (see Proposition 4.10). This is a key
property that relies on the modified construction process described above (and
is certainly not true in the setting of Section 2.2), and allows to avoid the
divergence in Section 2.2 caused by the factor $\mathfrak{D}^{n}$.

  * •

Vines and twists: The structure of _vines_ and _vine chains_ , which was
discovered in [18] (with a special case contained in [15]), is the main
obstacle in the analysis of non-regular couples. Such structures require a
delicate cancellation argument, where we combine the
$\mathcal{K}_{\mathcal{G}}$ terms for different gardens $\mathcal{G}$, that
are related by a special combinatorial operation called _twist_.

In this paper, all these arguments need to be adapted to the new layering
structure (consistency between twisting and layering structures is proved in
Proposition 7.19). We also need to introduce similar incoherency notions for
vines and vine chains (Definition 8.1), and establish similar gains for each
occurrence of incoherency, as with regular couples above, see Proposition 9.5.
The number of coherent vines and vine chains can again be controlled, see
Proposition 8.3.

  * •

Ladders: The _ladders_ are neutral objects observed in [15, 18] that do not
gain or lose any powers of $L$. As such, in this paper we only need to avoid
$\mathfrak{D}^{n}$ type loss in Section 2.2; just like for regular couples and
vines, we can define the incoherency notion for ladders (Definition 8.1), and
prove similar decay estimates and counting bounds for the given number of
occurrences of incoherency, see Propositions 8.2 and 10.4. Finally we need an
$L^{1}$ estimate for time integrals, similar to [15, 18] but adapted to the
layered case, see Proposition 10.5.

####  2.3.2. Structure of the paper

The rest of this paper is organized as follows. In Section 3 we introduce the
relevant notations (couples, gardens etc.), including the fundamental objects
of layered gardens and canonical layered gardens, and write down the main
ansatz. Then Sections 4–6 are devoted to the study of regular couples: Section
4 contains the combinatorial results, Section 5 contains the decay estimate
for incoherent regular couples, and Section 6 contains the asymptotics for the
coherent ones. In Sections 7–8 we define the notions of molecules, vines and
twists, and prove relevant combinatorial results for layerings of these
objects. In Section 9 we prove the main estimate for vines and vine chains.
Putting these together, in Sections 10–11 we perform the necessary reduction
steps to the molecule associated with a given garden $\mathcal{G}$, and apply
the blackbox algorithm in [15, 18] to complete the desired estimates for
$\mathcal{K}_{\mathcal{G}}$. Finally, in Section 12 we provide the estimates
for the linearization operator and the remainder term, and complete the proof
of Theorem 1.1. In the whole proof process we will need various auxiliary
lemmas (combinatorial, analytical, number theoretic and counting), which are
listed in Appendix A. Then Appendix B illustrates the failure of some physical
space $L^{1}$ bound in [48] (discussed in Section 1.4, Idea 2), and Appendix C
contains three tables collecting some of the important notations used in this
paper.

##  3\. Preparations for the proof

###  3.1. First reductions

Let $\delta$ be a small parameter to be fixed below, such that
$\tau_{*}=\mathfrak{D}\delta,\,\mathfrak{D}\in\mathbb{Z}$ (see Section 3.3.2).
We perform the first reduction for the equation (NLS) as follows. Let $u$ be a
solution to (NLS), and recall $\alpha=L^{-\gamma}$. Let $M=\fint|u|^{2}$ be
the conserved mass of $u$ (where $\fint$ takes the average on
$\mathbb{T}_{L}^{d}$), and define $v:=e^{-2iL^{-\gamma}Mt}\cdot u$, then $v$
satisfies the Wick ordered equation

$(i\partial_{t}-\Delta)v+L^{-\gamma}\bigg{(}|v|^{2}v-2\fint|v|^{2}\cdot
v\bigg{)}=0.$ (3.1)

By switching to Fourier space, rescaling in time and reverting the linear
Schrödinger flow, we define

$a_{k}(t)=e^{-\pi i\cdot\delta L^{2\gamma}|k|^{2}t}\cdot\widehat{v}(\delta
T_{\mathrm{kin}}\cdot t,k)$ (3.2)

with $\widehat{v}$ as in (1.1). Then $\textit{{a}}:=a_{k}(t)$, where
$t\in[0,\mathfrak{D}]$ and $k\in\mathbb{Z}_{L}^{d}$, will satisfy the equation

$\left\\{\begin{aligned}
\partial_{t}a_{k}&=\mathcal{C}_{+}(\textit{{a}},\overline{\textit{{a}}},\textit{{a}})_{k}(t),\\\
a_{k}(0)&=(a_{k})_{\mathrm{in}}=\sqrt{\varphi_{\mathrm{in}}(k)}\cdot\mathfrak{g}_{k},\end{aligned}\right.$
(3.3)

with the nonlinearity

$\mathcal{C}_{\zeta}(\textit{{f}},\textit{{g}},\textit{{h}})_{k}(t):=\frac{\delta}{2L^{d-\gamma}}\cdot(i\zeta)\sum_{k_{1}-k_{2}+k_{3}=k}\epsilon_{k_{1}k_{2}k_{3}}e^{\zeta\pi
i\cdot\delta
L^{2\gamma}\Omega(k_{1},k_{2},k_{3},k)t}f_{k_{1}}(t)g_{k_{2}}(t)h_{k_{3}}(t)$
(3.4)

for $\zeta\in\\{\pm\\}$. Here in (3.4) and below, the summation is taken over
$(k_{1},k_{2},k_{3})\in(\mathbb{Z}_{L}^{d})^{3}$, and

$\epsilon_{k_{1}k_{2}k_{3}}=\left\\{\begin{aligned} +&1,&&\mathrm{if\
}k_{2}\not\in\\{k_{1},k_{3}\\};\\\ -&1,&&\mathrm{if\ }k_{1}=k_{2}=k_{3};\\\
&0,&&\mathrm{otherwise},\end{aligned}\right.$ (3.5)

and the resonance factor (assuming $k_{1}-k_{2}+k_{3}=k$) is

$\Omega=\Omega(k_{1},k_{2},k_{3},k):=|k_{1}|^{2}-|k_{2}|^{2}+|k_{3}|^{2}-|k|^{2}=2\langle
k_{1}-k,k-k_{3}\rangle.$ (3.6)

Note that $\epsilon_{k_{1}k_{2}k_{3}}$ is always supported in the set

$\mathfrak{G}:=\big{\\{}(k_{1},k_{2},k_{3}):\mathrm{\ either\
}k_{2}\not\in\\{k_{1},k_{3}\\},\mathrm{\ or\ }k_{1}=k_{2}=k_{3}\big{\\}}.$
(3.7)

The rest of this paper is focused on the system (3.3)–(3.4) for a, with the
relevant terms defined in (3.5)–(3.6), in the time interval
$t\in[0,\mathfrak{D}]$.

###  3.2. Moments and cumulants

Let $a_{k}(t)$ be the (random) solution to (3.3)–(3.4). By gauge ($u\mapsto
e^{i\theta}u$) and space translation ($u(x)\mapsto u(x-x_{0})$) invariance it
is easy to see that, for any $k_{j}\in\mathbb{Z}_{L}^{d}$,
$t_{j}\in\mathbb{R}$ and $\zeta_{j}\in\\{\pm\\}$ we have

$\mathbb{E}\bigg{(}\prod_{j=1}^{r}a_{k_{j}}(t_{j})^{\zeta_{j}}\bigg{)}=0$
(3.8)

unless $\zeta_{1}+\cdots+\zeta_{r}=0$ (in particular $r$ is even) and
$\zeta_{1}k_{1}+\cdots+\zeta_{r}k_{r}=0$ (in particular $k_{1}=k_{2}=k$ when
$r=2$). Note that any event $\mathfrak{F}$ can be interpreted as a subset of
initial data $u_{\mathrm{in}}$ as in (DAT). We say an event $\mathfrak{F}$ has
gauge or space translation symmetry, if the corresponding subset is gauge or
space translation invariant. Now we recall the standard notion of cumulants
(see [48]):

######  Definition 3.1 (Cumulants [48]).

For any complex random variables $X_{j}\,(1\leq j\leq r)$, define the
_cumulant_

$\mathbb{K}(X_{1},\cdots,X_{r}):=\sum_{\mathcal{P}}(-1)^{|\mathcal{P}|-1}(|\mathcal{P}|-1)!\prod_{A\in\mathcal{P}}\mathbb{E}\bigg{(}\prod_{j\in
A}X_{j}\bigg{)},$ (3.9)

where the sum is taken over all partitions $\mathcal{P}$ of
$\\{1,\cdots,r\\}$. We then have the inversion formula (see for example [48])

$\mathbb{E}(X_{1}\cdots
X_{r}):=\sum_{\mathcal{P}}\prod_{A\in\mathcal{P}}\mathbb{K}\big{(}(X_{j})_{j\in
A}\big{)}.$ (3.10)

Note that, due to (3.8), if $X_{j}=a_{k_{j}}(t_{j})^{\zeta_{j}}$, then (3.9)
also vanishes unless $\zeta_{1}+\cdots+\zeta_{r}=0$ and
$\zeta_{1}k_{1}+\cdots+\zeta_{r}k_{r}=0$, and in both (3.9) and (3.10) we can
always assume that $(\zeta_{j}:j\in A)$ contains half $+$ and half $-$ for
each $A\in\mathcal{P}$. In addition, if all the moments and cumulants are
taken assuming a particular event $\mathfrak{F}$ (i.e. taken with an
additional factor $\mathbf{1}_{\mathfrak{F}}$) with the same gauge and space
translation symmetries, then all the above conclusions remain true.

###  3.3. Norms, parameters, and notations

We next define the necessary norms, fix the parameters and introduce the
relevant notations.

####  3.3.1. Norms

In the rest of this paper, we use $\widehat{\cdot}$ _only_ for the time
Fourier transform, defined as

$\widehat{u}(\lambda)=\int_{\mathbb{R}}u(t)e^{-2\pi i\lambda
t}\,\mathrm{d}t,\quad u(t)=\int_{\mathbb{R}}\widehat{u}(\lambda)e^{2\pi
i\lambda t}\,\mathrm{d}\lambda,$

and similarly for higher dimensional versions.

For $\theta\in\mathbb{R}$, integers $\Lambda_{1},\Lambda_{2}\geq 0$ and a
function $F=F(t_{1},\cdots,t_{r},k)$ on $\mathbb{R}^{r}\times\mathbb{R}^{d}$,
where $t_{j}\in\mathbb{R}$ are the time variables and $k\in\mathbb{R}^{d}$ is
the vector variable, define the
$\mathfrak{X}^{\theta,\Lambda_{1},\Lambda_{2}}$ norm

$\|F\|_{\mathfrak{X}^{\theta,\Lambda_{1},\Lambda_{2}}}=\max_{|\rho|\leq\Lambda_{1}}\int(\max(\langle\lambda_{1}\rangle,\cdots,\langle\lambda_{r}\rangle))^{\theta}\cdot\|\langle
k\rangle^{\Lambda_{2}}\partial_{k}^{\rho}\widehat{F}(\lambda_{1},\cdots,\lambda_{r},k)\|_{L_{k}^{\infty}}\,\mathrm{d}\lambda_{1}\cdots\mathrm{d}\lambda_{r}.$
(3.11)

In special cases where $F=F(t_{1},\cdots,t_{r})$ or $F=F(k)$, define
accordingly the norms

$\|F\|_{\mathfrak{X}^{\theta}}=\int(\max(\langle\lambda_{1}\rangle,\cdots,\langle\lambda_{r}\rangle))^{\theta}\cdot|\widehat{F}(\lambda_{1},\cdots,\lambda_{r})|\,\mathrm{d}\lambda_{1}\cdots\mathrm{d}\lambda_{r}$
(3.12)
$\|F\|_{\mathfrak{S}^{\Lambda_{1},\Lambda_{2}}}=\max_{|\rho|\leq\Lambda_{1}}\|\langle
k\rangle^{\Lambda_{2}}\partial_{k}^{\rho}F(k)\|_{L_{k}^{\infty}}.$ (3.13)

Suppose $F=F(t_{1},\cdots,t_{r},k)$ is a function on
$\mathcal{B}\times\mathbb{Z}_{L}^{d}$ (or any subset of $\mathbb{R}^{d}$
instead of $\mathbb{Z}_{L}^{d}$), where $\mathcal{B}\subset\mathbb{R}^{r}$ is
a set of time variables $(t_{j})$, then we may define the
$\mathfrak{X}^{\theta,\Lambda_{1},\Lambda_{2}}(\mathcal{B})$ norm of $F$ by

$\|F\|_{\mathfrak{X}^{\theta,\Lambda_{1},\Lambda_{2}}(\mathcal{B})}=\inf\big{\\{}\|G\|_{\mathfrak{X}^{\theta,\Lambda_{1},\Lambda_{2}}}:G|_{\mathcal{B}\times\mathbb{Z}_{L}^{d}}=F\big{\\}}.$
(3.14)

The same applies to norms $\mathfrak{X}^{\theta}$ (so we can define the
$\mathfrak{X}^{\theta}(\mathcal{B})$ norm) and
$\mathfrak{S}^{\Lambda_{1},\Lambda_{2}}$ (so it can be defined for functions
on $\mathbb{Z}_{L}^{d}$). In addition, if $\Lambda_{1}=\Lambda_{2}=0$, we can
define the $\mathfrak{X}^{\theta,0,0}$ and
$\mathfrak{X}^{\theta,0,0}(\mathcal{B})$ norms for functions $F$ depending on
multiple variables $k_{j}\,(1\leq j\leq q)$ with the same formula.

Finally, for any interval $\mathcal{B}$, integer $\Lambda\geq 0$ and any
function $\textit{{a}}=a_{k}(t)$ where $t\in\mathcal{B}$ and
$k\in\mathbb{Z}_{L}^{d}$, define the norm

$\|\textit{{a}}\|_{Z^{\Lambda}(\mathcal{B})}=\sup_{t\in\mathcal{B}}\sup_{k\in\mathbb{Z}_{L}^{d}}\langle
k\rangle^{\Lambda}|a_{k}(t)|.$ (3.15)

####  3.3.2. Parameters

First recall the local well-posedness result proved in [31]; for our purpose,
we state it below in the form of the (equivalent) blowup criterion.

######  Proposition 3.2.

Given any Schwartz initial data $\varphi_{\mathrm{in}}$, there exists a
maximal time $\tau_{\max}>0$ (may be $\infty$) such that (WKE) has a unique
solution $\varphi=\varphi(\tau,k)$ in Schwartz class, for
$\tau\in[0,\tau_{\max})$. Moreover we have the blowup criterion

$\lim_{\tau\to\tau_{\max}}\|\langle
k\rangle^{s}\varphi(\tau,k)\|_{L_{k}^{2}}=\infty,\quad\forall s>1/2.$ (3.16)

###### Proof.

See Proposition 2.4 of [31]. Preservation of Schwartz properties follows
easily from the form of the collision operator (COL). ∎

Throughout this paper, we will fix $\tau_{\mathrm{max}}$ as in Proposition
3.2, and fix $\tau_{*}\in(0,\tau_{\mathrm{max}})$ as in Theorem 1.1. We also
fix the solution $\varphi(\tau,k)$ to (WKE) as in Theorem 1.1 and Proposition
3.2. We will use the notation $C_{0}$ to denote arbitrary large constant that
depends only on $(d,\gamma)$, and fix $\eta$ that is small enough depending
only on $C_{0}$. Define

$\mathfrak{C}:=\sup_{\tau\in[0,\tau_{*}]}\|\varphi(\tau)\|_{\mathfrak{S}^{40d,40d}}<\infty,$
(3.17)

with $\mathfrak{S}^{40d,40d}$ defined as above, which depends only on
$\tau_{*}$ and the initial data $\varphi_{\mathrm{in}}$. We use $C_{1}$ to
denote arbitrary large constant that depends only on $(\tau_{*},\eta)$ and
$\mathfrak{C}$. Fix also $\nu$ that is small enough depending on $C_{1}$, and
$\delta$ that is small enough depending on $\nu$, such that

$\mathfrak{D}:=\frac{\tau_{*}}{\delta}\in\mathbb{Z}.$ (3.18)

We will use $C_{2}$ to denote arbitrary large constant that depends only on
$\mathfrak{D}$ and $\delta$ (and also $\varphi_{\mathrm{in}}$), and assume $L$
is large enough depending on $C_{2}$. For integer $0\leq p\leq\mathfrak{D}$,
define

$\theta_{p}:=\eta^{20}(1+2^{-p})\,(\mathrm{and\
}\theta=\theta_{\mathfrak{D}}\mathrm{\ in\ Theorem\
\ref{main}}),\quad\Lambda_{p}:=(40d)^{\mathfrak{D}-p+1},$ (3.19)

note that
$\theta_{p+1}<\frac{4\theta_{p+1}+\theta_{p}}{5}<\frac{3\theta_{p+1}+2\theta_{p}}{5}<\frac{2\theta_{p+1}+3\theta_{p}}{5}<\frac{\theta_{p+1}+4\theta_{p}}{5}<\theta_{p}$.
Define also $N_{p}$ and $R_{p}$ such that888This choice is just for
convenience; the same proof allows for any choice of $R_{\mathfrak{D}}$
between $(\log L)^{40d}$ and $L^{\nu^{2}(40d)^{-2\mathfrak{D}}}$.

$R_{\mathfrak{D}}:=\lfloor(\log L)^{40d}\rfloor;\quad
N_{p}=R_{p}^{40d}\,(0\leq p\leq\mathfrak{D}),\quad R_{p-1}=N_{p}^{40d}\,(1\leq
p\leq\mathfrak{D}).$ (3.20)

Finally, define

$\gamma_{0}:=\min(\gamma,1-\gamma),\quad\gamma_{1}:=\min(2\gamma,1,2(d-1)(1-\gamma)).$
(3.21)

####  3.3.3. Notations

We will use $X=\mathcal{O}_{j}(Y)$, $X\lesssim_{j}Y$ and $X\ll_{j}Y$ etc.,
where $j\in\\{0,1,2\\}$, to indicate that the implicit constants depend only
on $C_{j}$ (where $C_{j}$ are defined in Section 3.3.2). Note that by
definition we have $\nu\ll_{1}1$ and $\delta^{\nu}\ll_{1}1$. For multi-index
$\rho=(\rho_{1},\cdots,\rho_{m})$, denote $|\rho|=\rho_{1}+\cdots+\rho_{m}$
and $\rho!=(\rho_{1})!\cdots(\rho_{m})!$, etc. For an index set $A$, we use
the vector notation $\alpha[A]=(\alpha_{j})_{j\in A}$ and
$\mathrm{d}\alpha[A]=\prod_{j\in A}\mathrm{d}\alpha_{j}$, etc. The sign
$\zeta\in\\{\pm\\}$ is understood as $\pm 1$ in algebraic manipulations.
Denote $x^{j}$ as the coordinates of vector
$(x^{1},\cdots,x^{d})\in\mathbb{R}^{d}$, and $z^{+}=z$ for a complex number
$z$, and $z^{-}=\overline{z}$.

For $x\in\mathbb{R}$ fix the cutoff function

$\chi_{0}=\chi_{0}(x)=\left\\{\begin{aligned} &0,&&\mathrm{for\ }|x|\geq 1;\\\
&1,&&\mathrm{for\ }|x|\leq 1/2;\\\ &\mathrm{smooth\ in\ }[0,1],&&\mathrm{for\
}1/2\leq|x|\leq 1.\end{aligned}\right.$

For $x\in\mathbb{R}^{d}$ define
$\chi_{0}(x)=\chi_{0}(x^{1})\cdots\chi_{0}(x^{d})$, and
$\chi_{\infty}=1-\chi_{0}$. In general, we may use $\chi$ to denote other
cutoff functions with slightly different supports. These functions, as well as
the other cutoff functions, will be in Gevrey class $2$ (i.e. the $m$-th order
derivatives are bounded by $(2m)!$).

###  3.4. Combinatorial structures

We define the combinatorial structures that will be used throughout this
paper, including trees, gardens, couples and their decorations. For
illustrations of structures defined in [15, 16, 18] the reader may refer to
those papers; here we only provide illustration for the new structures
introduced in this paper.

####  3.4.1. Trees, gardens and couples

We start with the notion of trees in [15], which will be restricted to ternary
trees throughout this paper.

######  Definition 3.3 (Trees [15]).

A _ternary tree_ $\mathcal{T}$ (we will simply say a _tree_ below) is a rooted
tree where each non-leaf (or _branching_) node has exactly three children
nodes, which we shall distinguish as the _left_ , _mid_ and _right_ ones. A
node $\mathfrak{m}$ is a _descendant_ of a node $\mathfrak{n}$, or
$\mathfrak{n}$ is an _ancestor_ of $\mathfrak{m}$, if $\mathfrak{m}$ belongs
to the subtree rooted at $\mathfrak{n}$ (we allow
$\mathfrak{m}=\mathfrak{n}$). We say $\mathcal{T}$ is _trivial_ (and write
$\mathcal{T}=\bullet$) if it has only the root, in which case this root is
also viewed as a leaf.

We denote generic nodes by $\mathfrak{n}$, generic leaves by $\mathfrak{l}$,
the root by $\mathfrak{r}$, the set of leaves by $\mathcal{L}$ and the set of
branching nodes by $\mathcal{N}$. The _order_ of a tree $\mathcal{T}$ is
defined by $n(\mathcal{T})=|\mathcal{N}|$, so if $n(\mathcal{T})=n$ then
$|\mathcal{L}|=2n+1$ and $|\mathcal{T}|=3n+1$. Denote the parent node of a
non-root node $\mathfrak{n}$ by $\mathfrak{n}^{\mathrm{pr}}$.

A tree $\mathcal{T}$ may have sign $+$ or $-$. If its sign is fixed then we
decide the signs of its nodes as follows: the root $\mathfrak{r}$ has the same
sign as $\mathcal{T}$, and for any branching node
$\mathfrak{n}\in\mathcal{N}$, the signs of the three children nodes of
$\mathfrak{n}$ from left to right are $(\zeta,-\zeta,\zeta)$ if $\mathfrak{n}$
has sign $\zeta\in\\{\pm\\}$. Once the sign of $\mathcal{T}$ is fixed, we will
denote the sign of $\mathfrak{n}\in\mathcal{T}$ by $\zeta_{\mathfrak{n}}$.
Define
$\zeta(\mathcal{T})=\prod_{\mathfrak{n}\in\mathcal{N}}(i\zeta_{\mathfrak{n}})$.
We also define the conjugate $\overline{\mathcal{T}}$ of a tree $\mathcal{T}$
to be the same tree but with opposite sign.

Next we define gardens (and the special case of couples), which are
collections of trees with their leaves paired. The definition of couples first
appeared in [15], and general gardens in [16].

######  Definition 3.4 (Gardens and couples [15, 16]).

Given a sequence $(\zeta_{1},\cdots,\zeta_{2R})$, where
$\zeta_{j}\in\\{\pm\\}$ and exactly half of them are $+$, we define a _garden_
$\mathcal{G}$ of _signature_ $(\zeta_{1},\cdots,\zeta_{2R})$, to be an ordered
collection of trees $(\mathcal{T}_{1},\cdots,\mathcal{T}_{2R})$, such that
$\mathcal{T}_{j}$ has sign $\zeta_{j}$ for $1\leq j\leq 2R$, together with a
partition $\mathscr{P}$ of the set of leaves in all $\mathcal{T}_{j}$ into
two-element subsets (called _pairings_) such that the two paired leaves have
opposite signs. The _width_ of the garden is defined to be $2R$, which is
always an even number. The _order_ $n(\mathcal{G})$ of a garden $\mathcal{G}$
is the sum of orders of all $\mathcal{T}_{j}\,(1\leq j\leq 2R)$. We denote
$\mathcal{L}=\mathcal{L}_{1}\cup\cdots\cup\mathcal{L}_{2R}$ to be the set of
leaves and $\mathcal{N}=\mathcal{N}_{1}\cup\cdots\cup\mathcal{N}_{2R}$ to be
the set of branching nodes, where $\mathcal{L}_{j}$ and $\mathcal{N}_{j}$ are
the sets of leaves and branching nodes of $\mathcal{T}_{j}$, and define
$\zeta(\mathcal{G})=\prod_{\mathfrak{n}\in\mathcal{N}}(i\zeta_{\mathfrak{n}})$.

A garden of width $2$ (i.e. $R=1$) is called a _couple_ , and denoted by
$\mathcal{Q}$. A non-couple garden (i.e. $R\geq 2$) is called _proper_. For a
couple we will always assume $(\zeta_{1},\zeta_{2})=(+,-)$ and denote
$(\mathcal{T}_{1},\mathcal{T}_{2})$ by $(\mathcal{T}^{+},\mathcal{T}^{-})$. We
also define a _paired tree_ to be a tree $\mathcal{T}$ whose leaves are all
paired to each other according to the same pairing rule for gardens and
couples, except for _one unpaired leaf_ which is called the _lone leaf_. Note
that if $\mathcal{T}$ is a paired tree, then it automatically forms a couple
with the trivial tree $\bullet$. Define the conjugate of a couple
$\mathcal{Q}=(\mathcal{T}^{+},\mathcal{T}^{-})$ as
$\overline{\mathcal{Q}}=(\overline{\mathcal{T}^{-}},\overline{\mathcal{T}^{+}})$;
for a paired tree $\mathcal{T}$ we also define its conjugate as
$\overline{\mathcal{T}}$ with the same pairings, where
$\overline{\mathcal{T}}$ is as in Definition 3.3.

We say a garden $\mathcal{G}$ is _irreducible_ if there exists no proper
subset $A\subset\\{1,\cdots,2R\\}$ (i.e. $A\neq\varnothing$ and
$A\neq\\{1,\cdots,2R\\}$) such that the leaves of $\mathcal{T}_{j}\,(j\in A)$
are all paired with each other. Clearly any garden $\mathcal{G}$ can be
uniquely decomposed into irreducible gardens corresponding to a partition
$(A_{i})$ of $\\{1,\cdots,2R\\}$, where $|A_{i}|$ is even and $(\zeta_{j}:j\in
A_{i})$ contains half $+$ and half $-$ for each $i$. These are called the
_components_ of $\mathcal{G}$. We say $\mathcal{G}$ is a _multi-couple_ if all
its components are couples (i.e. have width $2$), and _trivial_ if each tree
$\mathcal{T}_{j}$ is trivial. Note that a trivial garden must be a multi-
couple formed by $R$ trivial couples.

Next we define decorations of couples and gardens (this is defined in [15] for
couples, and [16] for general gardens).

######  Definition 3.5 (Decorations [15, 16]).

For any tree $\mathcal{T}$, a _decoration_ $\mathscr{D}$ of $\mathcal{T}$ is a
mapping $k[\mathcal{T}]:\mathcal{T}\to\mathbb{Z}_{L}^{d}$ consisting of
$k_{\mathfrak{n}}$ for each node $\mathfrak{n}$ of $\mathcal{T}$, such that

$k_{\mathfrak{n}_{1}}-k_{\mathfrak{n}_{2}}+k_{\mathfrak{n}_{3}}=k_{\mathfrak{n}},\quad\mathrm{equivalently}\quad\zeta_{\mathfrak{n}_{1}}k_{\mathfrak{n}_{1}}+\zeta_{\mathfrak{n}_{2}}k_{\mathfrak{n}_{2}}+\zeta_{\mathfrak{n}_{3}}k_{\mathfrak{n}_{3}}=\zeta_{\mathfrak{n}}k_{\mathfrak{n}}$

for any branching node $\mathfrak{n}$ and its three children nodes
$(\mathfrak{n}_{1},\mathfrak{n}_{2},\mathfrak{n}_{3})$ from left to right
(recall $\zeta_{\mathfrak{n}}$ defined in Definition 3.3). Clearly a
decoration $\mathscr{D}$ is uniquely determined by the values of
$(k_{\mathfrak{l}})_{\mathfrak{l}\in\mathcal{L}}$. For
$k\in\mathbb{Z}_{L}^{d}$, we say $\mathscr{D}$ is a $k$-decoration if
$k_{\mathfrak{r}}=k$ for the root $\mathfrak{r}$.

For any garden $\mathcal{G}=(\mathcal{T}_{1},\cdots,\mathcal{T}_{2R})$, a
_decoration_ $\mathscr{I}$ of $\mathcal{G}$ is a mapping
$k[\mathcal{G}]:\mathcal{G}\to\mathbb{Z}_{L}^{d}$ consisting of
$k_{\mathfrak{n}}$ for each node $\mathfrak{n}$ of $\mathcal{G}$, such that
the restriction to each $\mathcal{T}_{j}$ is a decoration of
$\mathcal{T}_{j}$, and $k_{\mathfrak{l}}=k_{\mathfrak{l}^{\prime}}$ for each
leaf pair $(\mathfrak{l},\mathfrak{l}^{\prime})$ of $\mathcal{G}$. For any
$k_{j}\in\mathbb{Z}_{L}^{d}\,(1\leq j\leq 2R)$, we say $\mathscr{I}$ is a
$(k_{1},\cdots,k_{2R})$-decoration if $k_{\mathfrak{r}_{j}}=k_{j}$ where
$\mathfrak{r}_{j}$ is the root of the tree $\mathcal{T}_{j}$. Note that in
this case we necessarily have $\zeta_{1}k_{1}+\cdots+\zeta_{2R}k_{2R}=0$,
where $\zeta_{j}$ are as in Definition 3.4; in particular for couples we must
have $k_{1}=k_{2}:=k$, thus for any $k\in\mathbb{Z}_{L}^{d}$ we may also talk
about $k$-decorations for couples. We can define decorations of paired trees
in the same way (again requiring that
$k_{\mathfrak{l}}=k_{\mathfrak{l}^{\prime}}$ for each leaf pair
$(\mathfrak{l},\mathfrak{l}^{\prime})$); in this case we must have
$k_{\mathfrak{r}}=k_{\mathfrak{l}_{\mathrm{lo}}}:=k$ for the root
$\mathfrak{r}$ and lone leaf $\mathfrak{l}_{\mathrm{lo}}$, so we may also talk
about $k$-decorations of paired trees. Moreover, _for couples and paired
trees_ , we shall define $k$-decorations for _general vectors_
$k\in\mathbb{R}^{d}$, by requiring $k_{\mathfrak{n}}-k\in\mathbb{Z}_{L}^{d}$
instead of $k_{\mathfrak{n}}\in\mathbb{Z}_{L}^{d}$, and keeping all other
requirements.

For any decoration $\mathscr{D}$ or $\mathscr{I}$ defined above, we define the
coefficient

$\epsilon_{\mathscr{D}}\,(\mathrm{or\
}\epsilon_{\mathscr{I}})=\prod_{\mathfrak{n}\in\mathcal{N}}\epsilon_{k_{\mathfrak{n}_{1}}k_{\mathfrak{n}_{2}}k_{\mathfrak{n}_{3}}}$
(3.22)

where $(\mathfrak{n}_{1},\mathfrak{n}_{2},\mathfrak{n}_{3})$ is as above and
$\epsilon_{k_{1}k_{2}k_{3}}$ is as in (3.5); in the support of
$\epsilon_{\mathscr{D}}$ (or $\epsilon_{\mathscr{I}}$) we have
$(k_{\mathfrak{n}_{1}},k_{\mathfrak{n}_{2}},k_{\mathfrak{n}_{3}})\in\mathfrak{G}$
as in (3.7) for any $\mathfrak{n}\in\mathcal{N}$. Define the resonance factor
$\Omega_{\mathfrak{n}}$ for each $\mathfrak{n}\in\mathcal{N}$ by

$\Omega_{\mathfrak{n}}=\Omega_{\mathfrak{n}}(k_{\mathfrak{n}_{1}},k_{\mathfrak{n}_{2}},k_{\mathfrak{n}_{3}},k_{\mathfrak{n}})=|k_{\mathfrak{n}_{1}}|^{2}-|k_{\mathfrak{n}_{2}}|^{2}+|k_{\mathfrak{n}_{3}}|^{2}-|k_{\mathfrak{n}}|^{2}.$
(3.23)

Note that, for a $k$-decoration of couples and paired trees, if we translate
all the $k_{\mathfrak{n}}$, including $k$, by the same constant vector
$\overline{k}\in\mathbb{R}^{d}$ (which may not be in $\mathbb{Z}_{L}^{d}$),
then the decoration conditions, as well as the values of
$\Omega_{\mathfrak{n}}$, do not change.

####  3.4.2. Layerings

Now we introduce the key new structure of _layerings_ imposed on gardens (and
couples and paired trees), which corresponds to iterating the solution at
different time slices.

######  Definition 3.6 (Layered gardens).

A _layering_ of a garden $\mathcal{G}$ is a mapping
$\mathfrak{L}[\mathcal{G}]:\mathcal{G}\to\mathbb{Z}$ consisting of
$\mathfrak{L}_{\mathfrak{n}}$ for each node $\mathfrak{n}$ of $\mathcal{G}$,
such that
$0\leq\mathfrak{L}_{\mathfrak{n}^{\prime}}\leq\mathfrak{L}_{\mathfrak{n}}$
whenever $\mathfrak{n}^{\prime}$ is a child of $\mathfrak{n}$ and
$\mathfrak{L}_{\mathfrak{l}}=\mathfrak{L}_{\mathfrak{l}^{\prime}}$ whenever
$(\mathfrak{l},\mathfrak{l}^{\prime})$ are two paired leaves, see Figure 4. We
call $\mathfrak{L}_{\mathfrak{n}}$ the _layer_ of $\mathfrak{n}$. A garden
$\mathcal{G}$ with a layering fixed is called a _layered_ garden. We also
define layerings of paired trees in the same way. Moreover, define a _pre-
layering_ (of a garden or a paired tree) to be the same as layering except we
define $\mathfrak{L}_{\mathfrak{n}}$ only for _branching nodes_
$\mathfrak{n}$, and according to the same rules as above.

Figure 4. A layered garden, as in Definition 3.6. The signs of trees are
$(+,-,+,-)$ from left to right.

We will consider an important class of layered gardens, namely the _canonical
layered gardens_ , which includes the ones that occur in the actual inductive
scheme (see Proposition 3.9).

######  Definition 3.7 (Canonical layered gardens).

Let $p\geq 0$. We inductively define a collection $\mathscr{G}_{p}$ of
irreducible proper layered gardens (see Definition 3.4); gardens in
$\mathscr{G}_{p}$ will be called _canonical layered gardens of depth $p$_. Let
$\mathscr{G}_{0}=\varnothing$. Suppose $\mathscr{G}_{p}$ is defined for some
$p\geq 0$, to define $\mathscr{G}_{p+1}$ we consider the following process.

First choose any $2R\,(R\geq 2)$ trees $\mathcal{T}_{j}^{p}\,(1\leq j\leq 2R)$
with signs $\zeta_{j}$ as in Definition 3.4 (half $+$ and half $-$), and put
all their branching nodes in layer $p$. Next, divide the set of leaves of all
these trees into subsets $B_{i}$ such that the signs of leaves in each subset
are half $+$ and half $-$; we also require that there is no proper subset
$A\subset\\{1,\cdots,2R\\}$ such that the set of leaves of
$\mathcal{T}_{j}^{p}$ for all $j\in A$ equals the union of _some of the sets_
$B_{i}$. Then, we turn any _two-element_ subset $B_{i}$ into a pairing,
putting the involved leaves again in layer $p$. For any subset $B_{i}$ of _at
least four elements_ , we replace its leaves by an irreducible proper layered
garden in $\mathscr{G}_{p}$, such that the roots of the trees in this garden
(including the signs) are just the elements of $B_{i}$. This results in a big
proper layered garden of width $2R$, which is easily shown to be irreducible,
and we define $\mathscr{G}_{p+1}$ to consist of all the irreducible proper
layered gardens that can be obtained in this way, see Figure 5.

We also define, for $p\geq 0$, a collection $\mathscr{C}_{p+1}$ of layered
_couples_ , by choosing $R=1$ and following the same procedure in the above
construction (i.e. choosing two trees of signs $+$ and $-$, dividing the set
of leaves into pairings and subsets of at least four elements, and replacing
the latters by layered gardens in $\mathscr{G}_{p}$). Couples in
$\mathscr{C}_{p+1}$ are called _canonical layered couples_.

Figure 5. An illustration showing that the layered garden in Figure 4 is
canonical, following the inductive construction in Definition 3.7.

Finally, we define an important quantity $\mathcal{K}_{\mathcal{G}}$ for
gardens $\mathcal{G}$ (as well as $\mathcal{K}_{\mathcal{Q}}^{*}$ and
$\mathcal{K}_{\mathcal{T}}^{*}$ for couples $\mathcal{Q}$ and paired trees
$\mathcal{T}$), which is similar to the ones in [15, 16], but also takes into
account the layers. This quantity is the main subject of study in the proof of
the current paper.

######  Definition 3.8 (Definition of $\mathcal{K}_{\mathcal{G}}$,
$\mathcal{K}_{\mathcal{Q}}^{*}$ and $\mathcal{K}_{\mathcal{T}}^{*}$).

Fix an integer $p\geq 0$, let $\mathcal{G}$ be a layered garden consisting of
trees $\mathcal{T}_{j}\,(1\leq j\leq 2R)$ with sign $\zeta_{j}$, such that all
nodes of $\mathcal{G}$ are in layer at most $p$. Fix also the input functions
$F_{q}=F_{q}(k)$ for $0\leq q\leq p$, then for any
$k_{1},\cdots,k_{2R}\in\mathbb{Z}_{L}^{d}$ (satisfying
$\zeta_{1}k_{1}+\cdots+\zeta_{2R}k_{2R}=0$, where
$(\zeta_{1},\cdots,\zeta_{2R})$ is the signature of $\mathcal{G}$) and $t>p$,
we may define

$\mathcal{K}_{\mathcal{G}}=\mathcal{K}_{\mathcal{G}}(t,k_{1},\cdots,k_{2R})=\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n}\zeta(\mathcal{G})\sum_{\mathscr{I}}\epsilon_{\mathscr{I}}\int_{\mathcal{I}}\prod_{\mathfrak{n}\in\mathcal{N}}e^{\pi
i\zeta_{\mathfrak{n}}\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}}\cdot\prod_{\mathfrak{l}\in\mathcal{L}}^{(+)}F_{\mathfrak{L}_{\mathfrak{l}}}(k_{\mathfrak{l}}).$
(3.24)

Here in (3.24) $n$ is the order of $\mathcal{G}$, $\zeta(\mathcal{G})$ is as
in Definition 3.4 and $\epsilon_{\mathscr{I}}$ as in Definition 3.5. The sum
is taken over all $(k_{1},\cdots,k_{2R})$-decorations $\mathscr{I}$ of
$\mathcal{G}$, $\Omega_{\mathfrak{n}}$ is as in (3.23), and the last product
is taken over all $\mathfrak{l}\in\mathcal{L}$ with sign $+$. The time domain
$\mathcal{I}$ is defined by

$\mathcal{I}=\left\\{t[\mathcal{N}]:\mathfrak{L}_{\mathfrak{n}}<t_{\mathfrak{n}}<\mathfrak{L}_{\mathfrak{n}}+1\quad\mathrm{and}\quad
0<t_{\mathfrak{n}^{\prime}}<t_{\mathfrak{n}}<t,\mathrm{\ whenever\
\mathfrak{n}^{\prime}\ is\ a\ child\ of\ \mathfrak{n}}\right\\},$ (3.25)

where $\mathfrak{L}_{\mathfrak{n}}$ is the layer of $\mathfrak{n}$ as in
Definition 3.6. Note that the definition of $\mathcal{I}$ depends only on the
_pre-layering_ of $\mathcal{G}$.

Now we can define $\mathcal{K}_{\mathcal{Q}}$ for couples $\mathcal{Q}$ via
(3.24), in which case we can allow $k_{1}=k_{2}:=k$ to be _arbitrary vector in
$\mathbb{R}^{d}$, provided that the input functions $F_{r}=F_{r}(k)$ for
$0\leq r\leq p$ are defined on $\mathbb{R}^{d}$, and that the sum in (3.24) is
still taken over all $k$-decorations $\mathscr{I}$ of $\mathcal{Q}$ as in
Definition 3.5._ For later use, we will extend this definition to
$\mathcal{K}_{\mathcal{Q}}^{*}$, and also define similar expression
$\mathcal{K}_{\mathcal{T}}^{*}$ for paired trees $\mathcal{T}$, as follows.

First, let $\mathcal{Q}=(\mathcal{T}^{+},\mathcal{T}^{-})$ be a layered couple
such that the two roots are in layer at most $q$ and $q^{\prime}$ respectively
where $q,q^{\prime}\leq p$, and fix the input functions $F_{r}=F_{r}(k)$ for
$0\leq r\leq p$ (which are defined on $\mathbb{R}^{d}$ as stated above). Then
for all $k\in\mathbb{R}^{d}$ and $t>q$, $s>q^{\prime}$, we may define
$\mathcal{K}_{\mathcal{Q}}^{*}=\mathcal{K}_{\mathcal{Q}}^{*}(t,s,k)$, in the
same way as (3.24), except that the time domain is now

$\mathcal{I}^{*}=\left\\{t[\mathcal{N}]:\mathfrak{L}_{\mathfrak{n}}<t_{\mathfrak{n}}<\mathfrak{L}_{\mathfrak{n}}+1\quad\mathrm{and}\quad
0<t_{\mathfrak{n}^{\prime}}<t_{\mathfrak{n}},\mathrm{\ whenever\
\mathfrak{n}^{\prime}\ is\ a\ child\ of\ \mathfrak{n}},\right.\\\
\left.\mathrm{and}\quad t_{\mathfrak{n}}<t\ \mathrm{(or\ }s),\mathrm{\
whenever\ \mathfrak{n}\ is\ a\ node\ of\ \mathcal{T}^{+}\ (or\
\mathcal{T}^{-})}\right\\}.$ (3.26)

This gives an extension of the expression $\mathcal{K}_{\mathcal{Q}}$ in
(3.24) when $\mathcal{G}=\mathcal{Q}$ is a couple and
$k_{1}=k_{2}=k\in\mathbb{R}^{d}$, such that the latter corresponds to the
special case $t=s>p$.

Similarly, let $\mathcal{T}$ be a layered paired tree such that the root is in
layer at most $q$ and the lone leaf is in layer $q^{\prime}$ where
$q^{\prime}\leq q\leq p$, and fix the input functions $F_{r}=F_{r}(k)$ for
$0\leq r\leq p$ which are defined on $\mathbb{R}^{d}$. Then _for all
$k\in\mathbb{R}^{d}$ and $(t,s)$ satisfying $t>q$, $s<q^{\prime}+1$ and
$t>s$_, we may define
$\mathcal{K}_{\mathcal{T}}^{*}=\mathcal{K}_{\mathcal{T}}^{*}(t,s,k)$ in the
same way as (3.24), except that (i) the sum is taken over all $k$-decorations
$\mathscr{D}$ of $\mathcal{T}$ where again $k$ may not be in
$\mathbb{Z}_{L}^{d}$, (ii) the last product is taken over all leaves
$\mathfrak{l}\in\mathcal{L}$ with sign $+$ _other than the lone leaf
$\mathfrak{l}_{\mathrm{lo}}$_, and (iii) the time domain is now

$\mathcal{D}^{*}=\big{\\{}t[\mathcal{N}]:\mathfrak{L}_{\mathfrak{n}}<t_{\mathfrak{n}}<\mathfrak{L}_{\mathfrak{n}}+1\quad\mathrm{and}\quad
0<t_{\mathfrak{n}^{\prime}}<t_{\mathfrak{n}}<t,\mathrm{\ whenever\
\mathfrak{n}^{\prime}\ is\ a\ child\ of\ \mathfrak{n}},\\\ \mathrm{and}\quad
t_{\mathfrak{n}_{\mathrm{lo}}^{\mathrm{pr}}}>s,\mathrm{\ where\
\mathfrak{n}_{\mathrm{lo}}^{\mathrm{pr}}\ is\ the\ parent\ node\ of\ the\
lone\ leaf\ \mathfrak{l}_{\mathrm{lo}}}\big{\\}}.$ (3.27)

###  3.5. The inductive scheme

With all the above preparations, we can now describe the inductive scheme
which is the central component in the proof of Theorem 1.1.

Recall that we are considering the system (3.3)–(3.4) for
$t\in[0,\mathfrak{D}]$. Moreover, as the distinction between different $C_{j}$
in Section 3.3.2 will be important, we also recall it here:

$C_{0}\ll\eta^{-1}\ll C_{1}\ll\delta^{-1}\ll C_{2}.$

Here $C_{0}$ is any “absolute” constant depending only on $(d,\gamma)$, which
can be absorbed by $\eta$; $C_{1}$ is much larger than $\eta^{-1}$ and also
depends on the uniform bound $\mathfrak{C}$ of the solution $\varphi$ to (WKE)
defined in (3.17), but can still be absorbed by the small parameter $\delta$
indicating the length of each short interval; $C_{2}$ is the largest and may
depend on the number $\mathfrak{D}$ of intervals which may be much larger than
$\delta^{-1}$, but can still be absorbed by any quantity that increases with
$L$.

######  Proposition 3.9.

Recall the parameters defined in (3.20). For $0\leq p\leq\mathfrak{D}$ and
$2\leq R\leq R_{p}$, we can define an event $\mathfrak{F}_{p}$, functions
$\mathscr{R}_{p}=\mathscr{R}_{p}(k)$ and $\mathscr{E}_{p}=\mathscr{E}_{p}(k)$,
and
$\widetilde{\mathscr{E}_{p}}=\widetilde{\mathscr{E}_{p}}(\zeta_{1},\cdots,\zeta_{2R},k_{1},\cdots,k_{2R})$,
such that $\mathscr{R}_{p}(k)$ is real valued, and the followings hold for
each $0\leq p\leq\mathfrak{D}$:

  1. (1)

Assuming $\mathfrak{F}_{p}$, the system (3.3)–(3.4) has a unique solution
$\textit{{a}}=a_{k}(t)$ on $[0,p]$. We also have
$\mathfrak{F}_{p}\supset\mathfrak{F}_{p+1}$ and
$\mathbb{P}(\mathfrak{F}_{p})\geq 1-e^{-N_{p}}$. Moreover each
$\mathfrak{F}_{p}$ has gauge and space translation symmetry; the expectations
$\mathbb{E}$ and cumulants $\mathbb{K}$ below are taken assuming
$\mathfrak{F}_{p}$, and they come with subscript $p$.

  2. (2)

Let the cumulant $\mathbb{K}$ be defined as in (3.9), and $\mathbb{K}_{p}$ as
in (1). For any $k,\ell\in\mathbb{Z}_{L}^{d}$ we have

$\mathbb{K}_{p}(a_{k}(p),\overline{a_{\ell}(p)})=\mathbb{E}_{p}(a_{k}(p)\overline{a_{\ell}(p)})=\mathbf{1}_{k=\ell}\cdot\big{(}\varphi(\delta
p,k)+\mathscr{R}_{p}(k)+\mathscr{E}_{p}(k)\big{)},$ (3.28)

where $\varphi$ is the solution to (WKE), the sub-leading term
$\mathscr{R}_{p}(k)$ and remainder $\mathscr{E}_{p}(k)$ satisfy

$\|\mathscr{R}_{p}(k)\|_{\mathfrak{S}^{\Lambda_{p},\Lambda_{p}}}\leq
L^{-\theta_{p}},\quad|\mathscr{E}_{p}(k)|\leq\langle
k\rangle^{-\Lambda_{p}}\cdot e^{-N_{p}}.$ (3.29)

  3. (3)

For any $2\leq R\leq R_{p}$, signs $\zeta_{j}\,(1\leq j\leq 2R)$ as in
Definition 3.4, and $k_{j}\in\mathbb{Z}_{L}^{d}\,(1\leq j\leq 2R)$, we have

$\mathbb{K}_{p}\big{(}a_{k_{1}}(p)^{\zeta_{1}},\cdots,a_{k_{2R}}(p)^{\zeta_{2R}}\big{)}=\sum_{\mathcal{G}}\mathcal{K}_{\mathcal{G}}(p,k_{1},\cdots,k_{2R})+\widetilde{\mathscr{E}_{p}}.$
(3.30)

Here the sum is taken over all _canonical layered gardens_
$\mathcal{G}\in\mathscr{G}_{p}$ of signature $(\zeta_{1},\cdots,\zeta_{2R})$
and _order at most $N_{p}$_ (in particular all nodes of $\mathcal{G}$ are in
layer at most $p-1$), and $\mathcal{K}_{\mathcal{G}}$ is as in (3.24) in
Definition 3.8 with input functions $F_{q}(k)=\varphi(\delta
q,k)+\mathscr{R}_{q}(k)$ for $0\leq q\leq p-1$. The remainder
$\widetilde{\mathscr{E}_{p}}$ is supported in the set
$\zeta_{1}+\cdots+\zeta_{2R}=0$ and
$\zeta_{1}k_{1}+\cdots+\zeta_{2R}k_{2R}=0$, and satisfies that

$|\widetilde{\mathscr{E}_{p}}|\leq\big{(}\langle k_{1}\rangle\cdots\langle
k_{2R}\rangle\big{)}^{-\Lambda_{p}}\cdot e^{-N_{p}}.$ (3.31)

  4. (4)

In addition, for any $n\leq N_{p}$, consider the expression on the right hand
side of (3.30), but with the summation taken over all canonical layered
gardens $\mathcal{G}$ _of order $n$_ and without the remainder
$\widetilde{\mathscr{E}_{p}}$. Then this expression, denoted by
$\mathscr{K}_{p,n}$, satisfies that

$|\mathscr{K}_{p,n}|\leq\big{(}\langle k_{1}\rangle\cdots\langle
k_{2R}\rangle\big{)}^{-\Lambda_{p}}\cdot L^{-(R-1)/6}\cdot\delta^{\nu n}.$
(3.32)

###  3.6. The Duhamel expansion

The proof of Proposition 3.9 will occupy most of the rest parts of this paper.
Since the statements (1)–(4) are trivially true for $p=0$ (with
$\mathscr{R}_{0}=\mathscr{E}_{0}=\widetilde{\mathscr{E}_{0}}=0$ and
$\mathfrak{F}_{0}$ the trivial true event) due to the independence and
Gaussian assumption (DAT), we only need to justify the inductive step. Thus,
for the rest of this paper (except Section 3.7), we will fix $0\leq
p\leq\mathfrak{D}-1$ and assume (1)–(4) of Proposition 3.9 hold for any $0\leq
q\leq p$, and we just need to prove the same for $p+1$.

To achieve this, we need to expand the solution $(a_{k}(p+1))$ in terms of the
previous data $(a_{k}(p))$. Assuming the event $\mathfrak{F}_{p}$, for any
tree $\mathcal{T}$ and any $t>p$, define the expression

$(\mathcal{J}_{\mathcal{T}})_{k}(t)=\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n}\zeta(\mathcal{T})\sum_{\mathscr{D}}\epsilon_{\mathscr{D}}\int_{\mathcal{D}}\prod_{\mathfrak{n}\in\mathcal{N}}e^{\pi
i\zeta_{\mathfrak{n}}\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}}\cdot\prod_{\mathfrak{l}\in\mathcal{L}}a_{k_{\mathfrak{l}}}(p).$
(3.33)

Here in (3.33), $n$ is the order of $\mathcal{T}$, $\zeta(\mathcal{T})$ is as
in Definition 3.3 and $\epsilon_{\mathscr{D}}$ as in Definition 3.5. The sum
is taken over all $k$-decorations $\mathscr{D}$ of $\mathcal{T}$, and
$\Omega_{\mathfrak{n}}$ is as in (3.23). The integral domain $\mathcal{D}$ is
defined by

$\mathcal{D}=\left\\{t[\mathcal{N}]:p<t_{\mathfrak{n}^{\prime}}<t_{\mathfrak{n}}<t,\mathrm{\
whenever\ \mathfrak{n}^{\prime}\ is\ a\ child\ of\ \mathfrak{n}}\right\\}.$
(3.34)

Now, define

$(\mathcal{J}_{n})_{k}(t)=\sum_{n(\mathcal{T}^{+})=n}(\mathcal{J}_{\mathcal{T}^{+}})_{k}(t)$
(3.35)

where the sum is taken over all trees $\mathcal{T}^{+}$ of order $n$ and sign
$+$. Define the function $\textit{{b}}=b_{k}(t)$, where $t\in[p,p+1]$ and
$k\in\mathbb{Z}_{L}^{d}$, by

$a_{k}(t)=a_{k}^{\mathrm{tr}}(t)+b_{k}(t),\quad
a_{k}^{\mathrm{tr}}(t):=\sum_{n=0}^{N_{p+1}}(\mathcal{J}_{n})_{k}(t).$ (3.36)

It is then easy to see that the system (3.3)–(3.4) for $\textit{{a}}=a_{k}(t)$
on $[p,p+1]$ is equivalent to the following equation for b:

$\textit{{b}}=\mathscr{L}_{0}+\mathscr{L}_{1}\textit{{b}}+\mathscr{L}_{2}(\textit{{b}},\textit{{b}})+\mathscr{L}_{3}(\textit{{b}},\textit{{b}},\textit{{b}}),$
(3.37)

or equivalently

$\textit{{b}}=(1-\mathscr{L}_{1})^{-1}(\mathscr{L}_{0}+\mathscr{L}_{2}(\textit{{b}},\textit{{b}})+\mathscr{L}_{3}(\textit{{b}},\textit{{b}},\textit{{b}})),$
(3.38)

where

$(\mathscr{L}_{j}(\textit{{b}},\cdots,\textit{{b}}))_{k}(t)=\sum_{(j)}\int_{p}^{t}\mathcal{C}_{+}(\textit{{u}},\textit{{v}},\textit{{w}})_{k}(s)\,\mathrm{d}s.$
(3.39)

The sums in (3.39) are taken over $(\textit{{u}},\textit{{v}},\textit{{w}})$,
where each of them equals either b or $\mathcal{J}_{n}$ for some $0\leq n\leq
N_{p+1}$. Assume in the sum $\sum_{(j)}$ that exactly $j$ inputs in
$(\textit{{u}},\textit{{v}},\textit{{w}})$ equal b, and in the sum
$\sum_{(0)}$ we further require that
$(\textit{{u}},\textit{{v}},\textit{{w}})=(\mathcal{J}_{n_{1}},\mathcal{J}_{n_{2}},\mathcal{J}_{n_{3}})$
with $n_{1}+n_{2}+n_{3}\geq N_{p+1}$. Note that each $\mathscr{L}_{j}$ is a
$j$-multilinear form over $\mathbb{R}$.

To calculate the cumulants

$\mathbb{K}_{p+1}\big{(}a_{k_{1}}(p+1)^{\zeta_{1}},\cdots,a_{k_{2R}}(p+1)^{\zeta_{2R}}\big{)}$
(3.40)

in Proposition 3.9, we first calculate the contribution of truncated tree
terms, namely

$\mathbb{K}_{p}\big{(}a_{k_{1}}^{\mathrm{tr}}(p+1)^{\zeta_{1}},\cdots,a_{k_{2R}}^{\mathrm{tr}}(p+1)^{\zeta_{2R}}\big{)}.$
(3.41)

Now, by Lemma A.1, Definition 3.7, gauge and space translation symmetry of
$\mathfrak{F}_{p}$, and the induction hypothesis, we can write the cumulant
(3.41) as the sum of one explicit term, plus remainder terms that contain one
of $\mathscr{E}_{p}$ or $\widetilde{\mathscr{E}_{p}}$. For $R\geq 2$ this
explicit term equals

$\sum_{\mathcal{G}}\mathcal{K}_{\mathcal{G}}(p+1,k_{1},\cdots,k_{2R}),$

where $\mathcal{K}_{\mathcal{G}}$ is as in (3.24) in Definition 3.8 with input
functions $F_{q}(k)=\varphi(\delta q,k)+\mathscr{R}_{q}(k)$ for $0\leq q\leq
p$, and $\mathcal{G}$ runs over a specific collection
$\mathscr{G}_{p+1}^{\mathrm{tr}}$ of canonical layered gardens in
$\mathscr{G}_{p+1}$; for $R=1$ we should replace $\mathcal{G}$ by a couple
$\mathcal{Q}$ which runs over a specific collection
$\mathscr{C}_{p+1}^{\mathrm{tr}}$ of canonical layered couples in
$\mathscr{C}_{p+1}$. Here $\mathscr{G}_{p+1}^{\mathrm{tr}}$ and
$\mathscr{C}_{p+1}^{\mathrm{tr}}$ are defined as follows:

######  Definition 3.10.

For any canonical layered garden $\mathcal{G}\in\mathscr{G}_{p+1}$ (or couple
$\mathcal{Q}\in\mathscr{C}_{p+1}$), we define
$\mathcal{G}\in\mathscr{G}_{p+1}^{\mathrm{tr}}$ (or
$\mathcal{Q}\in\mathscr{C}_{p+1}^{\mathrm{tr}}$) if and only if, following the
construction process in Definition 3.7, we have that (i) each initial tree
$\mathcal{T}_{j}^{p}$ in Definition 3.7 has order at most $N_{p+1}$, and (ii)
each layered garden in $\mathscr{G}_{p}$ that replaces the leaves in some
subset $B_{i}$ has order at most $N_{p}$.

We illustrate the above expansion by a concrete example, which corresponds to
the canonical layered garden in Figure 5. Consider (3.41) with $p=2$ and
$R=2$, and $\zeta_{j}=(-1)^{j-1}$. Replace the four $a^{\mathrm{tr}}$ factors
in (3.41), from left to right, by $\mathcal{J}_{\mathcal{T}_{j}^{2}}$ for the
four trees $\mathcal{T}_{j}^{2}$ in Figure 5 (where $\mathcal{T}_{2}^{2}$ and
$\mathcal{T}_{4}^{2}$ have sign $-$, corresponding to
$\zeta_{2}=\zeta_{4}=-$). Note that the root of $\mathcal{T}_{j}^{2}$ is
decorated by $k_{j}$, and we denote the decoration of the nodes in
$\mathcal{T}_{j}^{2}$ by $k_{ji}\,(i\geq 1)$ with numbering from top to bottom
and left to right; similarly we denote the time variables associated with
$\mathcal{T}_{j}^{2}$ by $t_{j}^{\prime}<t_{j}$ etc. This then leads to the
expression

$\displaystyle\sum_{\begin{subarray}{c}k_{1}=k_{11}-k_{12}+k_{13}\\\
k_{12}=k_{14}-k_{15}+k_{16}\end{subarray}}\sum_{k_{2}=k_{21}-k_{22}+k_{23}}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{3}(-i)\cdot\epsilon_{k_{11}k_{12}k_{13}}\epsilon_{k_{14}k_{15}k_{16}}\epsilon_{k_{21}k_{22}k_{23}}$
$\displaystyle\qquad\,\,\times\int_{2<t_{1}^{\prime}<t_{1}<3}e^{\pi i\delta
L^{2\gamma}(|k_{12}|^{2}-|k_{14}|^{2}+|k_{15}|^{2}-|k_{16}|^{2})t_{1}^{\prime}}\cdot
e^{\pi i\delta
L^{2\gamma}(|k_{11}|^{2}-|k_{12}|^{2}+|k_{13}|^{2}-|k_{1}|^{2})t_{1}}\,\mathrm{d}t_{1}^{\prime}\mathrm{d}t_{1}$
$\displaystyle\qquad\,\,\times\int_{2<t_{2}<3}e^{\pi i\delta
L^{2\gamma}(|k_{2}|^{2}-|k_{21}|^{2}+|k_{22}|^{2}-|k_{23}|^{2})t_{2}}\,\mathrm{d}t_{2}$
$\displaystyle\qquad\,\,\times\mathbb{K}_{2}\big{(}a_{k_{11}}(2)\overline{a_{k_{14}}(2)}a_{k_{15}}(2)\overline{a_{k_{16}}(2)}a_{k_{13}}(2),\overline{a_{k_{21}}(2)}a_{k_{22}}(2)\overline{a_{k_{23}}(2)},a_{k_{3}}(2),\overline{a_{k_{4}}(2)}\big{)}.$
(3.42)

Now we apply Lemma A.1 to the cumulant expression in (3.42); corresponding to
the pairings between $\mathcal{T}_{j}^{2}$ in Figure 5, we select the term
corresponding to

$\mathbb{K}_{2}(a_{k_{11}}(2),\overline{a_{k_{16}}(2)})\cdot\mathbb{K}_{2}(a_{k_{15}}(2),\overline{a_{k_{23}}(2)})\cdot\mathbb{K}_{2}\big{(}\overline{a_{k_{14}}(2)},a_{k_{13}}(2),\overline{a_{k_{21}}(2)},a_{k_{22}}(2),a_{k_{3}}(2),\overline{a_{k_{4}}(2)}\big{)},$

which also determines the sets $B_{i}$ in Definition 3.7. By induction
hypothesis, the first two cumulants above can be replaced by
$\mathbf{1}_{k_{11}=k_{16}}\cdot F_{2}(k_{11})$ etc. (ignoring
$\mathscr{E}_{p}$ and $\widetilde{\mathscr{E}_{p}}$, same below); the last
cumulant is the sum of

$\mathcal{K}_{\mathcal{G}^{\prime}}(2,k_{14},k_{13},k_{21},k_{22},k_{3},k_{4})$

over all $\mathcal{G}^{\prime}\in\mathscr{G}_{2}$ (with signature
$(-,+,-,+,+,-)$ and an upper bound for $n(\mathcal{G}^{\prime})$). If we
select the garden $\mathcal{G}^{\prime}\in\mathscr{G}_{2}$ in Figure 5, we get
the factor

$\displaystyle\sum_{k_{21}=k_{24}-k_{25}+k_{26}}\sum_{k_{3}=k_{31}-k_{32}+k_{33}}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{2}(+1)\cdot\epsilon_{k_{24}k_{25}k_{26}}\epsilon_{k_{31}k_{32}k_{33}}$
$\displaystyle\qquad\,\,\times\int_{1<t_{2}^{\prime}<2}\int_{0<t_{3}<1}e^{\pi
i\delta
L^{2\gamma}(|k_{21}|^{2}-|k_{24}|^{2}+|k_{25}|^{2}-|k_{26}|^{2})t_{2}^{\prime}}\cdot
e^{\pi i\delta
L^{2\gamma}(|k_{31}|^{2}-|k_{32}|^{2}+|k_{33}|^{2}-|k_{3}|^{2})t_{3}}\,\mathrm{d}t_{2}^{\prime}\mathrm{d}t_{3}$
$\displaystyle\qquad\,\,\times\mathbf{1}_{k_{14}=k_{33}}F_{0}(k_{14})\cdot\mathbf{1}_{k_{13}=k_{26}}F_{1}(k_{13})\cdot\mathbf{1}_{k_{24}=k_{31}}F_{0}(k_{24})\cdot\mathbf{1}_{k_{25}=k_{4}}F_{1}(k_{25})\cdot\mathbf{1}_{k_{22}=k_{32}}F_{0}(k_{22}).$

Plugging this into (3.42), we get the total expression

$\displaystyle\sum_{\begin{subarray}{c}k_{1}=k_{11}-k_{12}+k_{13}\\\
k_{12}=k_{14}-k_{15}+k_{16}\end{subarray}}\sum_{\begin{subarray}{c}k_{2}=k_{21}-k_{22}+k_{23}\\\
k_{21}=k_{24}-k_{25}+k_{26}\end{subarray}}\sum_{k_{3}=k_{31}-k_{32}+k_{33}}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{5}(-i)\cdot\epsilon_{k_{11}k_{12}k_{13}}\epsilon_{k_{14}k_{15}k_{16}}\epsilon_{k_{21}k_{22}k_{23}}$
$\displaystyle\qquad\,\,\times\int_{2<t_{1}^{\prime}<t_{1}<3}e^{\pi i\delta
L^{2\gamma}(|k_{12}|^{2}-|k_{14}|^{2}+|k_{15}|^{2}-|k_{16}|^{2})t_{1}^{\prime}}\cdot
e^{\pi i\delta
L^{2\gamma}(|k_{11}|^{2}-|k_{12}|^{2}+|k_{13}|^{2}-|k_{1}|^{2})t_{1}}\,\mathrm{d}t_{1}^{\prime}\mathrm{d}t_{1}$
$\displaystyle\qquad\,\,\times\int_{1<t_{2}^{\prime}<2}\int_{2<t_{2}<3}e^{\pi
i\delta
L^{2\gamma}(|k_{2}|^{2}-|k_{21}|^{2}+|k_{22}|^{2}-|k_{23}|^{2})t_{2}}\cdot
e^{\pi i\delta
L^{2\gamma}(|k_{21}|^{2}-|k_{24}|^{2}+|k_{25}|^{2}-|k_{26}|^{2})t_{2}^{\prime}}\,\mathrm{d}t_{2}^{\prime}\mathrm{d}t_{2}$
$\displaystyle\qquad\,\,\times\epsilon_{k_{24}k_{25}k_{26}}\epsilon_{k_{31}k_{32}k_{33}}\int_{0<t_{3}<1}e^{\pi
i\delta
L^{2\gamma}(|k_{31}|^{2}-|k_{32}|^{2}+|k_{33}|^{2}-|k_{3}|^{2})t_{3}}\,\mathrm{d}t_{3}\cdot\mathbf{1}_{k_{11}=k_{16}}F_{2}(k_{11})\cdot\mathbf{1}_{k_{15}=k_{23}}F_{2}(K_{15})$
$\displaystyle\qquad\,\,\times\mathbf{1}_{k_{14}=k_{33}}F_{0}(k_{14})\cdot\mathbf{1}_{k_{13}=k_{26}}F_{1}(k_{13})\cdot\mathbf{1}_{k_{24}=k_{31}}F_{0}(k_{24})\cdot\mathbf{1}_{k_{25}=k_{4}}F_{1}(k_{25})\cdot\mathbf{1}_{k_{22}=k_{32}}F_{0}(k_{22}),$

which is exactly $\mathcal{K}_{\mathcal{G}}(3,k_{1},k_{2},k_{3},k_{4})$ for
the canonical layered garden $\mathcal{G}$ illustrated in Figure 5.

Now we recall that, due to the smallness of $\delta$ relative to
$\mathfrak{C}$ defined in (3.17), the solution $\varphi((p+1)\delta,k)$ to the
wave kinetic equation (WKE) admits a power series expansion in terms of the
data $\varphi(p\delta,k)$, namely, for $k\in\mathbb{R}^{d}$:

$\varphi((p+1)\delta,k)=\sum_{n=0}^{\infty}\mathcal{U}_{n}(p+1,k),$ (3.43)

where $\mathcal{U}_{n}(t,k)$ is defined for $t\in[p,p+1]$ such that

$\mathcal{U}_{0}(t,k)=\varphi(p\delta,k),\quad\mathcal{U}_{n}(t,k)=\delta\sum_{n_{1}+n_{2}+n_{3}=n-1}\int_{p}^{t}\mathcal{K}(\mathcal{U}_{n_{1}}(t^{\prime}),\mathcal{U}_{n_{2}}(t^{\prime}),\mathcal{U}_{n_{3}}(t^{\prime}))(k)\,\mathrm{d}t^{\prime},$
(3.44)

where $\mathcal{K}$ is defined as in (COL). It is easy to see that

$\sup_{t\in[p,p+1]}\|\mathcal{U}_{n}(t,\cdot)\|_{\mathfrak{S}^{4\Lambda_{p+1},4\Lambda_{p+1}}}\leq
C_{2}\cdot(C_{1}\delta)^{n}$ (3.45)

for any $n\geq 0$.

We can now state the main estimates needed in the inductive step. They are
listed in the following propositions:

######  Proposition 3.11.

Given any $1\leq R\leq 10R_{p+1}$ and $n\leq N_{p+1}^{4}$, consider the
expression

$\mathscr{K}_{p+1,n}^{\mathrm{tr}}(\zeta_{1},\cdots,\zeta_{2R},k_{1},\cdots,k_{2R}):=\sum_{\mathcal{G}}\mathcal{K}_{\mathcal{G}}(p+1,k_{1},\cdots,k_{2R}).$
(3.46)

Here, for $R\geq 2$, $\mathcal{G}$ runs over all canonical layered gardens in
$\mathscr{G}_{p+1}^{\mathrm{tr}}$ of signature $(\zeta_{1},\cdots,\zeta_{2R})$
and order $n$; for $R=1$ we replace $\mathcal{G}$ by a couple $\mathcal{Q}$
which runs over all canonical layered couples in
$\mathscr{C}_{p+1}^{\mathrm{tr}}$ of order $n$, where
$\mathscr{G}_{p+1}^{\mathrm{tr}}$ and $\mathscr{C}_{p+1}^{\mathrm{tr}}$ are
defined in Definition 3.10. Note that when $n\leq N_{p+1}^{4}$, the
requirement (ii) in Definition 3.10, namely each layered garden in
$\mathscr{G}_{p}$ that replaces the leaves in some subset $B_{i}$ has order at
most $N_{p}$, is redundant. Then we have that:

  1. (1)

If $R\geq 2$, then for any choices of $(\zeta_{1},\cdots,\zeta_{2R})$, we have

$|\mathscr{K}_{p+1,n}^{\mathrm{tr}}|\leq\big{(}\langle
k_{1}\rangle\cdots\langle k_{2R}\rangle\big{)}^{-4\Lambda_{p+1}}\cdot
L^{-(R-1)/4}\cdot\delta^{2\nu n}.$ (3.47)

  2. (2)

If $R=1$, the expression $\mathscr{K}_{p+1,n}^{\mathrm{tr}}(+,-,k_{1},k_{2})$
is nonzero only when $k_{1}=k_{2}:=k$. In this case, we have

$\displaystyle\mathscr{K}_{p+1,2n}^{\mathrm{tr}}(+,-,k,k)$
$\displaystyle=\mathcal{U}_{n}(p+1,k)+\mathscr{R}_{p+1,2n}(k),$ (3.48)
$\displaystyle\mathscr{K}_{p+1,2n+1}^{\mathrm{tr}}(+,-,k,k)$
$\displaystyle=\mathscr{R}_{p+1,2n+1}(k),$

where $\mathcal{U}_{n}$ is defined in (3.44); moreover for any $n\leq
N_{p+1}^{4}$ we have

$\|\mathscr{R}_{p+1,n}(k)\|_{\mathfrak{S}^{4\Lambda_{p+1},4\Lambda_{p+1}}}\leq
L^{-(4\theta_{p+1}+\theta_{p})/5}\cdot\delta^{2\nu n}.$ (3.49)

######  Proposition 3.12.

There exists an event $\mathfrak{F}_{p+1}\subset\mathfrak{F}_{p}$, which has
gauge and space translation symmetry, such that
$\mathbb{P}(\mathfrak{F}_{p}\backslash\mathfrak{F}_{p+1})\leq e^{-4N_{p+1}}$.
Moreover, assuming $\mathfrak{F}_{p+1}$, we have the following estimates:

  1. (1)

For the term $a_{k}^{\mathrm{tr}}(t)$, and each $k\in\mathbb{Z}_{L}^{d}$ and
$t\in[p,p+1]$, we have

$|(a_{k}^{\mathrm{tr}})(t)|\leq\langle k\rangle^{-(3/2)\Lambda_{p+1}}\cdot
e^{3N_{p+1}}.$ (3.50)

  2. (2)

For the term $\mathscr{L}_{0}$ defined in (3.39), and each
$k\in\mathbb{Z}_{L}^{d}$ and $t\in[p,p+1]$, we have

$|(\mathscr{L}_{0})_{k}(t)|\leq\langle k\rangle^{-(3/2)\Lambda_{p+1}}\cdot
e^{-20N_{p+1}}.$ (3.51)

  3. (3)

The $\mathbb{R}$-linear operator $\mathscr{L}_{1}$ defined in (3.39) satisfies
that $1-\mathscr{L}_{1}$ is invertible in the space
$Z:=Z^{(5/4)\Lambda_{p+1}}([p,p+1])$ (see (3.15) for definition), and that

$\|(1-\mathscr{L}_{1})^{-1}\|_{Z\to Z}\leq e^{3N_{p+1}}.$ (3.52)

Finally, assuming (3.50)–(3.52), the mapping

$\textit{{b}}\mapsto(1-\mathscr{L}_{1})^{-1}(\mathscr{L}_{0}+\mathscr{L}_{2}(\textit{{b}},\textit{{b}})+\mathscr{L}_{3}(\textit{{b}},\textit{{b}},\textit{{b}}))$
(3.53)

is a contraction mapping from the metric space $\\{\textit{{b}}\in
Z:\|\textit{{b}}\|_{Z}\leq e^{-10N_{p+1}}\\}$ to itself.

###  3.7. Proof of Proposition 3.9 and Theorem 1.1

Assuming Propositions 3.11–3.12, we can now prove Proposition 3.9 and Theorem
1.1.

###### Proof of Proposition 3.9.

As pointed out in Section 3.6, we only need to prove the statements (1)–(4) in
Proposition 3.9 for $p+1$. Let the event $\mathfrak{F}_{p+1}$ be defined as in
Proposition 3.12. Then, assuming $\mathfrak{F}_{p+1}$, by the contraction
mapping property of (3.53), the equation (3.37) will have a unique solution
$\textit{{b}}=b_{k}(t)$, which means that $\textit{{a}}=a_{k}(t)$ as in (3.36)
solves (3.3)–(3.4) on $[p,p+1]$. As
$\mathfrak{F}_{p+1}\subset\mathfrak{F}_{p}$, we get a unique solution
$\textit{{a}}=a_{k}(t)$ to (3.3)–(3.4) on $[0,p+1]$, and

$\mathbb{P}(\mathfrak{F}_{p+1})=\mathbb{P}(\mathfrak{F}_{p})-\mathbb{P}(\mathfrak{F}_{p}\backslash\mathfrak{F}_{p+1})\geq
1-e^{-N_{p}}-e^{-4N_{p+1}}\geq 1-e^{-N_{p+1}},$

which proves statement (1).

Next, for any $1\leq R\leq R_{p+1}$ and any
$(\zeta_{1},\cdots,\zeta_{2R},k_{1},\cdots,k_{2R})$, consider the cumulant
(3.40). By gauge and space translation symmetry, we know it satisfies the
support conditions required in (2) and (3) of Proposition 3.9. As
$a_{k}(t)=a_{k}^{\mathrm{tr}}(t)+b_{k}(t)$, we shall first consider the
contribution (3.41). Following the discussions in Section 3.6, we expand each
$a_{k}^{\mathrm{tr}}$ into tree terms of order at most $N_{p+1}$ (see (3.36)),
apply Lemma A.1 and replace each resulting cumulant term using the induction
hypothesis; note that these cumulants have degree at most $2R^{\prime}\leq
10R_{p+1}N_{p+1}\leq R_{p}/10$ (recall $R_{p}=N_{p+1}^{40d}$ and
$N_{p+1}=R_{p+1}^{40d}$). Using Definition 3.7 and the induction hypothesis,
we get

$(\ref{cmattr})=\sum_{\mathcal{G}}\mathcal{K}_{\mathcal{G}}(p+1,k_{1},\cdots,k_{2R})+\mathscr{E}_{p+1}^{(1)},$
(3.54)

where $\mathcal{K}_{\mathcal{G}}$ is as in (3.24) in Definition 3.8 with input
functions $F_{q}(k)=\varphi(\delta q,k)+\mathscr{R}_{q}(k)$ for $0\leq q\leq
p$, and $\mathcal{G}$ runs over $\mathscr{G}_{p+1}^{\mathrm{tr}}$ as defined
in Definition 3.10; for $R=1$ we should replace $\mathcal{G}$ by a couple
$\mathcal{Q}$ which runs over $\mathscr{C}_{p+1}^{\mathrm{tr}}$.

The error term $\mathscr{E}_{p+1}^{(1)}$ in (3.54) contains those
contributions where at least one of the resulting cumulant terms is replaced
by $\mathscr{E}_{p}$ or $\widetilde{\mathscr{E}_{p}}$. By (3.29) and (3.31)
for $p$, these terms decay like $e^{-N_{p}}$, where recall that
$N_{p}=R_{p}^{40d}$; the other cumulant terms can be estimated using (3.32)
(by summing over all $n\leq N_{p}$), and they are simply bounded by
$L^{-(R-1)/6}\leq 1$. Since the whole expression involves the summation of at
most $10R_{p+1}N_{p+1}$ vector variables
$k_{\mathfrak{l}}\in\mathbb{Z}_{L}^{d}$, and the negative powers $\langle
k_{j}\rangle^{-\Lambda_{p}}$ in (3.29), (3.31) and (3.32) allows us to
restrict each $k_{\mathfrak{l}}$ to a fixed unit ball using summability, we
conclude that

$|\mathscr{E}_{p+1}^{(1)}|\leq\big{(}\langle k_{1}\rangle\cdots\langle
k_{2R}\rangle\big{)}^{-\Lambda_{p}/4}\cdot L^{20R_{p+1}N_{p+1}}\cdot
e^{-N_{p}}$

pointwise, where the negative power of $\max(\langle
k_{1}\rangle,\cdots,\langle k_{2R}\rangle)$ also comes from the $\langle
k_{\mathfrak{l}}\rangle^{-\Lambda_{p}}$ powers above. This is clearly
sufficient for the bounds (3.29) and (3.31) for $p+1$.

Next consider the sum of the $\mathcal{K}_{\mathcal{G}}$ terms in (3.54), and
assume $n(\mathcal{G})=n$. If $n>N_{p+1}^{4}$, then in the process of
replacing each cumulant term using the induction hypothesis as described
above, one of these cumulants must be replaced by a
$\mathcal{K}_{\mathcal{G}^{\prime}}$ term as in (3.30) for some
$\mathcal{G}^{\prime}$ with $n(\mathcal{G}^{\prime})\geq N_{p+1}^{2}$ (because
the number of $\mathcal{G}^{\prime}$ involved at most $10R_{p+1}N_{p+1}$,
while the sum of all these $n(\mathcal{G}^{\prime})$ is $n\geq N_{p+1}^{4}$).
Since we are also summing in $\mathcal{G}^{\prime}$ with the value of
$n(\mathcal{G}^{\prime})$ fixed, using (3.32) for $p$, we can estimate the sum
of $\mathcal{K}_{\mathcal{G}^{\prime}}$ by $e^{-N_{p+1}^{2}}$ and the product
of other cumulants simply by $1$ as above, with the same decay factors
$\langle k_{\mathfrak{l}}\rangle^{-\Lambda_{p}}$. This contribution, denoted
by $\mathscr{E}_{p+1}^{(2)}$, thensatisfies that

$|\mathscr{E}_{p+1}^{(2)}|\leq\big{(}\langle k_{1}\rangle\cdots\langle
k_{2R}\rangle\big{)}^{-\Lambda_{p}/4}\cdot L^{20R_{p+1}N_{p+1}}\cdot
e^{-N_{p+1}^{2}}$

pointwise, which is also sufficient for (3.29) and (3.31) for $p+1$.

Moreover, if $N_{p+1}<n\leq N_{p+1}^{4}$, denote this contribution by
$\mathscr{E}_{p+1}^{(3)}$, then we can use (3.47) and (3.49) from Proposition
3.11, to bound it by

$|\mathscr{E}_{p+1}^{(3)}|\leq\big{(}\langle k_{1}\rangle\cdots\langle
k_{2R}\rangle\big{)}^{-(3/2)\Lambda_{p+1}}\cdot e^{-10N_{p+1}},$

which is again sufficient for (3.29) and (3.31) for $p+1$. This finally allows
us to restrict to $n\leq N_{p+1}$, in which case both requirements (i) and
(ii) in the definition of $\mathscr{G}_{p+1}^{\mathrm{tr}}$ and
$\mathscr{C}_{p+1}^{\mathrm{tr}}$ in Definition 3.10 become redundant.
Therefore, in (3.54) we can just sum over all
$\mathcal{G}\in\mathscr{G}_{p+1}$ (or $\mathcal{Q}\in\mathscr{C}_{p+1}$) that
has order at most $N_{p+1}$, which matches the description of Proposition 3.9
for $p+1$. Apart from the difference between (3.40) and (3.41), this already
proves statement (3) of Proposition 3.9; moreover statement (2) follows from
(3.48)–(3.49) of Proposition 3.11 together with (3.43) and (3.45) (to get real
valued $\mathscr{R}_{p+1}$ we simply take real part), and statement (4)
follows from (3.47) of Proposition 3.11.

Now it remains to consider the difference between (3.40) and (3.41), and prove
that it belongs to an error term. Recall the definition of $\mathbb{K}_{p+1}$
and $\mathbb{K}_{p}$ in Proposition 3.9 and that
$a_{k}(t)=a_{k}^{\mathrm{tr}}(t)+b_{k}(t)$, we know that this difference
consists of two terms, denoted by $\mathscr{E}_{p+1}^{(4)}$ and
$\mathscr{E}_{p+1}^{(5)}$. First $\mathscr{E}_{p+1}^{(4)}$ contains terms of

$\mathbb{K}_{p+1}\big{(}c_{k_{1}}(p+1)^{\zeta_{1}},\cdots,c_{k_{2R}}(p+1)^{\zeta_{2R}}\big{)}$

where each $c_{k}(t)$ is either $a_{k}^{\mathrm{tr}}(t)$ or $b_{k}(t)$ but
with at least one factor being $b_{k}(t)$; note that, assuming the event
$\mathfrak{F}_{p+1}$, we always have $\|b_{k}(t)\|_{Z}\leq e^{-10N_{p+1}}$ by
Proposition 3.12, where $Z:=Z^{(5/4)\Lambda_{p+1}([p,p+1])}$. Now there are at
most $2^{2R}$ choice for $c_{j}$ which can be easily absorbed, and using (3.9)
we can reduce the cumulants to the corresponding moments with a loss of at
most $(C_{0}R_{p+1})!$. Then, by using Cauchy-Schwartz for each moment, we can
control $\mathscr{E}_{p+1}^{(4)}$ by terms of form

$|\mathscr{E}_{p+1}^{(4)}|\leq(C_{0}R_{p+1})!\cdot
e^{-10N_{p+1}}\prod_{j\not\in A}\langle
k_{j}\rangle^{-(5/4)\Lambda_{p+1}}\cdot\bigg{[}\mathbb{E}\bigg{(}\mathbf{1}_{\mathfrak{F}_{p}}\prod_{j\in
A}|a_{k_{j}}^{\mathrm{tr}}(p+1)|^{2}\bigg{)}\bigg{]}^{1/2}$

for some $A\subset\\{1,\cdots,2R\\}$. The last moment expression can again be
rewritten, using (3.10), in terms of cumulants of
$\\{a_{k_{j}}^{\mathrm{tr}}(p+1)\\}$ and
$\\{\overline{a_{k_{j}}^{\mathrm{tr}}(p+1)}\\}$ at another loss of at most
$(C_{0}R_{p+1})!$, while each of these cumulants has the form (3.40) (with $R$
replaced by $2R$, which does not matter). By repeating all the above
arguments, we can bound

$\mathbb{E}\bigg{(}\mathbf{1}_{\mathfrak{F}_{p}}\prod_{j\in
A}|a_{k_{j}}^{\mathrm{tr}}(p+1)|^{2}\bigg{)}\leq(C_{0}R_{p+1})!\cdot\prod_{j\in
A}\langle k_{j}\rangle^{-3\Lambda_{p+1}},$ (3.55)

and putting together we get the bound for $\mathscr{E}_{p+1}^{(4)}$ that is
sufficient for (3.29) and (3.31) for $p+1$, noticing that
$N_{p+1}=R_{p+1}^{40d}$. Finally, the term $\mathscr{E}_{p+1}^{(5)}$ equals
the difference

$(\mathbb{K}_{p+1}-\mathbb{K}_{p})\big{(}a_{k_{1}}^{\mathrm{tr}}(p+1)^{\zeta_{1}},\cdots,a_{k_{2R}}^{\mathrm{tr}}(p+1)^{\zeta_{2R}}\big{)}.$

By (3.9), we can write this as a sum of products of moments, with at least one
moment taken assuming the event
$\mathfrak{F}_{p}\backslash\mathfrak{F}_{p+1}$. By using another Cauchy-
Schwartz, we can bound each such moment by (3.55), with at least one extra
factor $\mathbb{P}(\mathfrak{F}_{p}\backslash\mathfrak{F}_{p+1})^{1/2}$. This
is again sufficient for (3.29) and (3.31) for $p+1$, since
$\mathbb{P}(\mathfrak{F}_{p}\backslash\mathfrak{F}_{p+1})\leq e^{-4N_{p+1}}$
by (3.12). This proves Proposition 3.9. ∎

###### Proof of Theorem 1.1.

Let $t=T_{\mathrm{kin}}\cdot\tau$ where $0<\tau\leq\tau_{*}$. Since
$\tau_{*}=\mathfrak{D}\cdot\delta$, we may assume $\tau=t^{\prime}\delta$ with
$t^{\prime}\in[p,p+1]$ for some $0\leq p<\mathfrak{D}$. First assume
$t^{\prime}=p$, then Proposition 3.9 already implies

$\mathbb{E}\big{(}\mathbf{1}_{\mathfrak{F}_{p}}|\widehat{u}(t,k)|^{2}\big{)}=\varphi(\tau,k)+\mathcal{O}_{0}(L^{-\theta_{p}})$

uniformly in $p$ and $k$, so we only need to consider the expectation with
$\mathbf{1}_{\mathfrak{F}\backslash\mathfrak{F}_{p}}$, where
$\mathfrak{F}\supset\mathfrak{F}_{p}$ is the event that (NLS) has a smooth
solution on $[0,T]$ (so
$\mathbb{P}(\mathfrak{F})\geq\mathbb{P}(\mathfrak{F}_{p})\geq\mathbb{P}(\mathfrak{F}_{\mathfrak{D}})\geq
1-e^{-(\log L)^{20d}}$). By mass conservation for (NLS), we have

$\mathbb{E}\big{(}\mathbf{1}_{\mathfrak{F}\backslash\mathfrak{F}_{p}}|\widehat{u}(t,k)|^{2}\big{)}\leq\sum_{k}\mathbb{E}\big{(}\mathbf{1}_{\mathfrak{F}\backslash\mathfrak{F}_{p}}|\widehat{u_{\mathrm{in}}}(k)|^{2}\big{)}\leq
2\sum_{k}\varphi_{\mathrm{in}}(k)\cdot\mathbb{P}(\mathfrak{F}\backslash\mathfrak{F}_{p})^{1/2}\leq
C_{0}L^{d}\cdot\mathbb{P}(\mathfrak{F}_{p}^{c})\to 0$

uniformly in $p$ and $k$, which then proves (1.2).

In general, suppose $t^{\prime}\in[p,p+1]$. We use Proposition 3.9 for $p$,
then, assuming $\mathfrak{F}_{p+1}$, we repeat the arguments applied in the
inductive step (which is used to prove Proposition 3.9 for $p+1$), but apply
them to rescaled time $t^{\prime}$ instead of $p+1$. It is clear from the
proof that none of the ingredients is affected by this change, and the result
follows in the same way. This proves Theorem 1.1. ∎

The rest of this paper is devoted to the proof of Propositions 3.11 and 3.12.
Recall that we have fixed $p\in[0,\mathfrak{D}-1]$ as in Section 3.6; this
value will be kept the same throughout the proof.

##  4\. Layered regular objects I: Combinatorics

###  4.1. Regular objects and structure theorem

First recall the definition and basic properties of _regular couples_ and
_regular trees_ in [15], which are of fundamental importance in both [15] and
the current paper.

######  Definition 4.1 (Regular couples and regular trees [15]).

Define a _$(1,1)$ -mini-couple_ to be a couple of order $2$ formed by two
trees of order $1$ with no siblings paired. It has two possibilities indicated
by codes $00$ and $01$, see Figure 6. Define a _mini tree_ to be a paired tree
of order $2$, again with no siblings paired. It has six possibilities
indicated by codes $10,11,20,21,30,31$, see Figure 7.

For any couple $\mathcal{Q}$ we can define two operations: operation A where a
leaf pair is replaced by a $(1,1)$-mini-couple, and operation B where a node
is replaced by a mini tree, see Figure 8. Then, we define a couple
$\mathcal{Q}$ to be _regular_ if it can be formed, starting from the trivial
couple, by operations A and B. We also define a paired tree $\mathcal{T}$ to
be a _regular tree_ , if $\mathcal{T}$ forms a regular couple with the trivial
tree $\bullet$. Clearly the order of any regular couple and regular tree must
be even.

Figure 6. A $(1,1)$-mini-couple as in Definition 4.1; the codes $00$ and $01$
indicate the two different possibilities. Figure 7. A mini-tree as in
Definition 4.1; the codes $10,11,20,21,30,31$ indicate the six different
possibilities. Figure 8. Operations $A$ and $B$ as in Definition 4.1; there
are different possibilities depending on the structure of the $(1,1)$-mini-
couple and mini-tree, see Figures 6 and 7. The ends $X$ and $Y$ here represent
the rest of the couple, which is unaffected by the operation.

We also need the notion of _sub-couples_ and _sub-paired-trees_ , which are
intuitively clear, and are made precise in the following definition.

######  Definition 4.2 (Embedded objects).

Let $\mathcal{G}$ be a garden, for nodes $\mathfrak{n}$ and
$\mathfrak{n}^{\prime}$, if the leaves in the two trees rooted at
$\mathfrak{n}$ and $\mathfrak{n}^{\prime}$ are all paired to each other, then
these two trees form a couple $\mathcal{Q}$, called a _sub-couple_ of
$\mathcal{G}$; we also say $\mathcal{Q}$ is _embedded in_ $\mathcal{G}$.
Similarly, if $\mathfrak{n}^{\prime}$ is a descendant of $\mathfrak{n}$, and
all the leaves that are descendants of $\mathfrak{n}$ but not descendants of
$\mathfrak{n}^{\prime}$ are paired to each other, then the tree rooted at
$\mathfrak{n}$ becomes a paired tree $\mathcal{T}$ after replacing the subtree
rooted at $\mathfrak{n}^{\prime}$ by a single (lone) leaf. In this case we
also call $\mathcal{T}$ a _sub-paired-tree_ and say $\mathcal{T}$ is _embedded
in_ $\mathcal{G}$. Also, starting from a garden $\mathcal{G}$, when we say we
“replace” a leaf pair by a couple, or a node by a paired tree (such as in
Definition 4.1), then this couple or paired tree will be embedded in the
resulting new garden $\mathcal{G}_{1}$. Conversely, we may “collapse” an
embedded couple or paired tree into a leaf pair or a node.

Before introducing the notion of layers, we first recall the definition of
regular chains in [15], as well as two structure theorems: first one for
regular couples and regular trees, and second one for general gardens. These
results are proved in [15] and [16].

######  Definition 4.3 (Regular chains [15]).

Given $m\geq 0$, consider a partition $\mathcal{P}$ of $\\{1,\cdots,2m\\}$
into $m$ pairwise disjoint two-element subsets (or pairs). We say this
partition $\mathcal{P}$ is _legal_ if there do not exist $a<b<c<d$ such that
$\\{a,c\\}\in\mathcal{P}$ and $\\{b,d\\}\in\mathcal{P}$; below we will always
assume $a<b$ and order these pairs $\\{a,b\\}$ in the increasing order of $a$.
Now, for any legal partition $\mathcal{P}$, we define a _regular chain_ to be
a paired tree of $2m$ branching nodes $\mathfrak{n}_{j}\,(1\leq j\leq 2m)$,
such that $\mathfrak{n}_{j+1}$ is a child of $\mathfrak{n}_{j}$ for $1\leq
j\leq 2m-1$, and some specified child of $\mathfrak{n}_{2m}$ is the lone leaf.
Moreover, for any $\\{a,b\\}\in\mathcal{P}$, the two remaining children nodes
of $\mathfrak{n}_{a}$ are paired with the two remaining children nodes of
$\mathfrak{n}_{b}$ with no siblings paired. Define also a _dominant chain_ to
be a regular chain where $\mathcal{P}=\\{\\{1,2\\},\cdots,\\{2m-1,2m\\}\\}$.

######  Proposition 4.4 (Structure theorem for regular objects).

Let $\mathcal{Q}$ be a nontrivial regular couple. Then exactly one of the two
cases happen:

  1. (a)

There exist unique regular couples $\mathcal{Q}_{j}\,(1\leq j\leq 3)$ such