# Phase transition for discrete nonlinear Schrödinger equation in three and
higher dimensions

Partha S. Dey  ,  Kay Kirkpatrick  and  Kesav Krishnan  Department of
Mathematics, University of Illinois Urbana-Champaign, 1409 W Green Street,
Urbana, Illinois 61801  $\\{$psdey, kkirkpat<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We analyze the thermodynamics of the focusing discrete nonlinear Schrödinger
equation in dimensions $d\geqslant 3$ with general nonlinearity $p>1$ and
under a model with two parameters, representing inverse temperature and
strength of the nonlinearity, respectively. We prove the existence of limiting
free energy and analyze the phase diagram for general $d,p$. We also prove the
existence of a continuous phase transition curve that divides the parametric
plane into two regions involving the appearance or non-appearance of solitons.
Appropriate upper and lower bounds for the curve are constructed that match
the result in [CK12] in a one-sided asymptotic limit. We also look at the
typical behavior of a function chosen from the Gibbs measure for certain parts
of the phase diagram.

###### Key words and phrases:

Nonlinear Schrödinger Equation, Invariant Measure, Solitons, Free energy,
Dispersive Equations

###### Contents

  1. 1 Introduction
  2. 2 Main Results
  3. 3 Background, Literature Review and Heuristics
  4. 4 Gaussian Free Field
  5. 5 Soliton Solutions and Minimal Energy
  6. 6 Convergence of the Free Energy
  7. 7 Analysis of Phases
  8. 8 Discussion on the Typical Function in the Subcritical Phase
  9. 9 Proofs of Main Theorems
  10. 10 Discussion and Further Questions

##  1\. Introduction

Nonlinear Schrödinger (NLS) equations have a fundamental physical importance.
They arise in descriptions of a multitude of classical and quantum phenomena,
examples include nonlinear optics [CLS03], Bose-Einstein condensation [BK04]
and even the complex dynamics of DNA [P04]. A close cousin of the NLS, the
Gross Pitaevski equation was recently used to describe a theory of dark matter
[KMW]. The focusing NLS is of significant mathematical interest due to the
competition between the dispersive character of the linear part of the
equation and the nonlinearity. A striking consequence is the formation of
solitons, localized solutions preserved in time. Additionaly, the NLS has
algebraic structure; it admits a Hamiltonian description and several conserved
quantities. In fact, in dimension one and for non-linearity $p=3$, the NLS is
completely integrable, _i.e.,_ it can be described in terms of a Lax pair
[LP68]. All this being said, the behavior of the focusing NLS is particularly
challenging to understand in higher dimensions and it is this situation that
we aim to address. We first discuss the continuum focusing nonlinear
Schrödinger Equation (NLS).

Let $\psi(t,x)$ be a complex-valued function of time $t$ and spatial variable
$x\in\mathds{R}^{d}$. We say that
$\psi:[0,\infty)\times\mathds{R}^{d}\to\mathds{C}$ satisfies the continuum
focusing NLS with power non-linearity $p>1$ if

$\displaystyle{\mathrm{i}\mkern
2.0mu}\partial_{t}\psi=-\Delta\psi-|\psi|^{p-1}\psi\text{ for all }t,x.$ (1)

The continuum Hamiltonian functional is given by

$\displaystyle
H_{c}(\psi):=\int_{\mathds{R}^{d}}|\nabla\psi|^{2}dx-\frac{2}{p+1}\int_{\mathds{R}^{d}}|\psi|^{p+1}dx.$
(2)

Formally (1) may be rewritten via the variation of the Hamiltonian as

$\displaystyle\frac{d}{dt}\psi={\mathrm{i}\mkern
2.0mu}\frac{\delta}{\delta\psi^{*}}H_{c}(\psi).$

Given the Hamiltonian structure, it is reasonable to address questions
regarding well-posedness and asymptotic behavior via the construction of
invariant measures for the flow. This approach has a rich history, and we will
survey the results about the existence of solutions and invariant measures in
Section 3. There is a significant obstacle to applying the standard method of
construction to the continuum equation in three dimensions and higher, which
we will also address in Section 3. Essentially, the natural candidate is not
normalizable due to spatial regularity issues (see [LRS88]). One way around
this obstacle is to consider a spatial discretization and study the discrete
NLS instead (see [CK12, C14]). For the spatial dimension, we fix an integer
$d\geqslant 3$. Let $\mathbb{T}_{n}^{d}$ be the $d$-dimensional discrete torus
with vertex set indexed by

$\displaystyle V=V_{n}=[n]^{d}:=\\{0,2,\ldots,n-1\\}^{d}$

of size $N=n^{d}$ and edge set $E=E_{n}$. We will denote the $d-$dimensional
integer lattice as $\mathds{Z}^{d}$. We take $h$ to be the spacing between two
neighboring vertices in either case. The discrete nearest neighbor Laplacian
acting on $\ell^{2}(G)$ with spacing $h$ and $G=\mathds{T}^{d}_{n}$ or
$\mathds{Z}^{d}$ (the case under consideration will always be specified as
required) is defined as

$\displaystyle\Delta_{h}\psi_{{\boldsymbol{x}}}:=\frac{1}{h^{2}}\sum_{\boldsymbol{y}\sim{\boldsymbol{x}}}(\psi_{{\boldsymbol{x}}}-\psi_{\boldsymbol{y}}),$
(3)

where $\boldsymbol{y}\sim{\boldsymbol{x}}$ denotes the sum over all nearest
neighbors $\boldsymbol{y}$ of ${\boldsymbol{x}}$ and
$\psi=(\psi_{{\boldsymbol{x}}})_{{\boldsymbol{x}}\in V}\in\ell^{2}(G)$. We may
regard the parameter $1/h$ to be the coupling strength of the lattice. The
focusing Discrete Nonlinear Schrödinger (DNLS) equation on $G$ with
nonlinearity $p$ is defined as coupled system of ODEs with
$(\psi_{{\boldsymbol{x}}}(t))_{{\boldsymbol{x}}\in V},t>0$ satisfying

$\displaystyle{\mathrm{i}\mkern
2.0mu}\frac{d}{dt}\psi_{{\boldsymbol{x}}}(t)=-\Delta_{h}\psi_{{\boldsymbol{x}}}(t)-|\psi_{{\boldsymbol{x}}}(t)|^{p-1}\psi_{{\boldsymbol{x}}}(t),\qquad{\boldsymbol{x}}\in
V,t>0.$ (4)

Equation (4) admits a global solution for $\ell^{2}$ initial data for both
cases of $G$. Like the continuum equation, it may be cast into a Hamiltonian
form. The discrete Hamiltonian associated with the focusing DNLS is

$\displaystyle\begin{split}\mathcal{H}_{h}(\psi)&=h^{d-2}\sum_{({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})\in
E}\left|\psi_{{\boldsymbol{x}}}-\psi_{{\boldsymbol{x}^{\prime}}}\right|^{2}-\frac{2}{p+1}\cdot
h^{d}\sum_{x\in G}|\psi_{{\boldsymbol{x}}}|^{p+1}\\\
&={h^{d-2}}\left\|\nabla\psi\right\|_{2}^{2}-\frac{2}{p+1}\cdot
h^{d}\left\|\psi\right\|_{p+1}^{p+1}.\end{split}$ (5)

Up to scaling by a constant, (5) is the discrete analog of (2). When defined
on the discrete torus, the phase space of the equation is isomorphic to
$\mathds{C}^{N}$, which crucially is finite-dimensional and has a natural
volume form. The volume form is preserved under the flow of the equation via
Liouville’s theorem. As a consequence of Noether’s theorem, the invariance of
the Hamiltonian with respect to multiplication of
$(\psi_{{\boldsymbol{x}}})_{{\boldsymbol{x}}\in V}$ by a constant phase
implies that the $\ell^{2}$ norm is a conserved by the dynamics. We refer to
the $\ell^{2}$ norm as the mass. We immediately obtain an invariant Gibbs
probability measure for the dynamics, defined on $\mathds{C}^{V}$ with
probability density of $(\psi_{{\boldsymbol{x}}})_{{\boldsymbol{x}}\in V}$
proportional to

$\displaystyle
e^{-\beta\mathcal{H}_{h}(\psi)}\cdot\mathds{1}_{\\{h^{d}\left\|\psi\right\|^{2}_{2}\leqslant
B\\}}\,\mathbf{d}\psi$ (6)

with $\beta,B>0$. For simplicity, we express the volume element as

$\displaystyle\mathbf{d}\psi:=\prod_{{\boldsymbol{x}}\in
V}d\Re{\psi_{{\boldsymbol{x}}}}\,d\Im{\psi}_{{\boldsymbol{x}}}.$

The relationship between $B$, $\beta$, $h$ and $N$ governs the scaling
behavior of this measure. In this article, we will work with the following
version of the problem.

######  Definition 1.1 (The model).

Let $d\geqslant 3,p>1$ be fixed. Given a positive real number $\nu>0$, we
consider the Hamiltonian

$\displaystyle
H_{\nu,N}(\psi):=\left\|\nabla\psi\right\|_{2}^{2}-\left(\frac{\nu}{N}\right)^{(p-1)/2}\cdot\frac{2}{p+1}\left\|\psi\right\|_{p+1}^{p+1}.$
(7)

for $\psi:\mathds{T}^{d}_{n}\to\mathds{C}$. With this Hamiltonian, we obtain a
Gibbs measure of the form

$\displaystyle\mu_{N}^{\theta,\nu}(\psi)=\frac{1}{Z_{N}(\theta,\nu)}e^{-\theta\mathcal{H}_{\nu,N}(\psi)}\cdot\mathds{1}_{\\{\left\|\psi\right\|_{2}^{2}\leqslant
N\\}}\mathbf{d}\psi$ (8)

where $\theta>0$ is the inverse temperature and

$\displaystyle
Z_{N}(\theta,\nu):=\int_{\mathds{C}^{V}}e^{-\theta\mathcal{H}_{\nu,N}(\psi)}\cdot\mathds{1}_{\\{\left\|\psi\right\|_{2}^{2}\leqslant
N\\}}\mathbf{d}\psi$ (9)

is the partition function.

In terms of the original measure (6), this corresponds to choosing our
parameters to obey

$\displaystyle\begin{split}\theta&=\beta B\cdot\frac{1}{Nh^{2}}\text{ and
}\nu=B\cdot h^{-d+4/(p-1)};\\\ \text{or, equivalently }\beta&=\theta/\nu\cdot
Nh^{-(d-2)+4/(p-1)}=\theta/\nu\cdot Nh^{-(d-2)(p-p_{e})/(p-1)},\\\ \text{ and
}B&=\nu\cdot h^{d-4/(p-1)}=\nu\cdot h^{d(p-p_{m})/(p-1)},\end{split}$ (10)

where $p_{m}:=1+4/d<p_{e}:=1+4/(d-2)$ are the mass critical and energy
critical threshold, respectively. When $h$ is of constant order, we get
$B=\Theta(1)$ whereas $\beta=\Theta(N)$. However, when $h\to 0$, we have $B\to
0$ or $\infty$ according as $p>p_{m}$ or $p<p_{m}$; and $\beta\ll N$ or $\gg
N$ according as $p>p_{e}$ or $p<p_{e}$. We will elaborate further on the
scaling in Section 3.5. It suffices to say here that the dependence in $N$ is
chosen such that asymptotically the linear and nonlinear parts of the energy
contribute on the same scale.

##  2\. Main Results

In order to state our results, we need to introduce two functions, $I$ and $W$
defined on $(0,\infty)$. They correspond to nonlinear and linear contributions
to the free energy respectively, as will be explained.

######  Definition 2.1 (Soliton with given mass).

Recall the discrete Hamiltonian introduced in (5), with $h=1$ and defined on
$\ell^{2}(\mathds{Z}^{d})$. We define the minimum energy at mass $a$ as

$\displaystyle I(a):=\inf_{\left\|\psi\right\|_{2}^{2}=a}\mathcal{H}(\psi).$
(11)

This is the energy of a soliton of mass $a$, when it exists. We will elaborate
more on this in Section 5. A standard result in the theory of DLNS states that
the minimizer is attained whenever $I(a)<0$. Moreover, there exists
$R_{p}=R_{p,d}\geqslant 0$ such that $I(a)=0$ iff $a\leqslant R_{p}$ (see
[WEI99]). It can also be shown that $I(a)\geqslant 0$ implies that $I(a)=0$.

As for the function $W$, let $\Delta$ denote the graph Laplacian on the
discrete torus. Let $\phi^{\boldsymbol{0}}$ denote the constant
$\boldsymbol{1}$ vector, which spans $\ker(\Delta)$, and $\Delta^{\perp}$ be
the restriction of $\Delta$ on
$\mathds{C}^{N}/\mathrm{span}\\{\phi^{\boldsymbol{0}}\\}$. We define

$\displaystyle K(y)=\lim_{N\to\infty}\frac{1}{N}\log\det(y-\Delta^{\perp}),\
y\in[0,\infty).$ (12)

In Section 4, we will show the convergence of the limit and give a more useful
expression for $K(y)$.

######  Definition 2.2 (Free field energy with given mass).

Let $K$ be as in (12). We define $W:(0,\infty)\to(0,\infty)$ as

$\displaystyle W(b):=\inf_{y\,:\,K^{\prime}(y)\leqslant b}(K(y)-yb).$ (13)

In Lemma 4.7 we will show that $W$ is a decreasing convex function. Moreover,
it is the limiting free energy of the Gaussian Free Field conditioned to have
mass $b$ and will be explicitly demonstrated in Section 4. There is a
$d-$dependent constant $C_{d}$ (see (26)) such that $W(b)=K(0)$ for all
$b\geqslant C_{d}$.

Throughout the rest of the paper, we will fix the spatial dimension
$d\geqslant 3$ and unless otherwise specified, the non-linearity will be fixed
$p>1$.

###  2.1. Free energy limit

Let $W(b)$ denote the limiting mean free energy of the Gaussian Free Field
conditioned to have mass $b$, as given in equation (13) and $I(a)$ denote the
minimum energy for the Hamiltonian (5) on $\mathds{Z}^{d}$ at mass $a$, as
given in equation (11). Our first main result is the following.

######  Theorem 2.3 (Convergence of free energy).

Let $Z_{N}(\theta,\nu)$ be as in (9). We have,

$\displaystyle\left|\frac{1}{N}\log
Z_{N}(\theta,\nu)-F(\theta,\nu)\right|\leqslant O(N^{-2(d-2)/{3d}}),$

where

$\displaystyle
F(\theta,\nu):=\log\frac{\pi}{\theta}-\min_{0<a<1}\left(W(\theta(1-a))+\frac{\theta}{\nu}I(a\nu)\right).$
(14)

Essentially, Theorem 2.3 states that asymptotically the mass of a typical
function may be divided into two parts:

  1. (1)

Structured part having mass $\approx an$, localized to a region of size
$\Theta(1)$, function values of $\psi_{x}$ are of order $\sqrt{N}$ and
contributes $\exp(-N\theta\nu^{-1}I(\nu a)+o(N))$ to the free energy.

  2. (2)

Random part having mass $\approx bN$ with $b\leqslant 1-a$, maximum value of
$|\psi_{x}|^{2}$ is $o(N)$, Gaussian fluctuation dominates the typical
behavior and contribution to free energy is given by the integral

$\displaystyle\int_{\sum_{x}|\psi_{x}|^{2}\ \approx\
bN}\exp\left(-\theta\left\|\nabla\psi\right\|_{2}^{2}\right)\mathbf{d}\psi$
$\displaystyle=(1/\theta)^{N}\cdot\int_{\sum_{x}|\psi_{x}|^{2}\ \approx\
b\theta N}\exp\left(-\left\|\nabla\psi\right\|_{2}^{2}\right)\mathbf{d}\psi$
$\displaystyle=(\pi/\theta)^{N}\cdot\exp(-NW(b\theta)+o(N)).$

Optimizing over $a,b,a+b\leqslant 1$ should give us the scaling behavior for
$Z_{N}(\theta,\nu)$. We prove that this is indeed the case.

###  2.2. Phase transition curve

With the behavior of the free energy established, the question moves towards
the behavior of the phases, which we characterize by the mass fraction $a$
allocated to the soliton portion. We will prove in Lemma 4.7 that $W$ is
continuous and differentiable. In particular, if the minimizer is attained at
$a=a_{\star}\in(0,1)$, then by differentiability of $W,I$, we get the relation
$W^{\prime}(\theta(1-a_{\star}))+I^{\prime}(a_{\star}\nu)=0.$ However, we have
no explicit formula for $I^{\prime}$ and only an implicit formula for
$W^{\prime}$, making it difficult to utilize this relation. We will still be
able to characterize the phase diagram in Section 2.2. We define

$\displaystyle\mathscr{M}(\theta,\nu):=\operatorname{argmin}_{0\leqslant
a\leqslant 1}\left(W(\theta(1-a))+\frac{\theta}{\nu}I(a\nu)\right)$ (15)

as the set of minimizers for the variation formula. This is a compact set by
continuity of $W$ and $I$. We further define

$\displaystyle a_{\star}(\theta,\nu):=\min\mathscr{M}(\theta,\nu)$ (16)

as the smallest $a$ attaining the global minima for (14). We define

$\displaystyle\mathscr{S}:=\\{(\theta,\nu)\in(0,\infty)^{2}:a_{\star}(\theta,\nu)>0\\}.$
(17)

as the open region in the $(\theta,\nu)$ phase-space having non-zero solitonic
contribution and

$\mathscr{D}:=\text{int}((0,\infty)^{2}\setminus\mathscr{S}),$

as the open region in the $(\theta,\nu)$ phase-space having zero solitonic
contribution. Note that

$\displaystyle\mathscr{S}$
$\displaystyle=\\{(\theta,\nu)\in(0,\infty)^{2}:\min_{0\leqslant a\leqslant
1}\left((W(\theta(1-a))-W(\theta))/\theta+I(a\nu)/\nu\right)<0\\}$
$\displaystyle=\bigcup_{\varepsilon>0,\
0<a<1}\left\\{(\theta,\nu)\in(0,\infty)^{2}:(W(\theta(1-a))-W(\theta))/\theta+I(a\nu)/\nu<-\varepsilon\right\\}$

is an open set by continuity of the map
$(\theta,\nu)\mapsto(W(\theta(1-a))-W(\theta))/\theta+I(a\nu)/\nu$ for fixed
$a$. The following result characterizes the phase transition. Define the
function

$\displaystyle\xi_{p}(t):=\inf_{0<a<1}\frac{-\ln(1-a)}{a^{(p+1)/2}+t}\text{
for }t\geqslant 0.$ (18)

See Figure 1 for a plot of $\xi_{p}(0)$.

Figure 1. Plot of $p$ vs. $\xi_{p}(0)$.

######  Theorem 2.4 (Existence of Phase transition curve).

We have the following results.

  1. a)

For $\nu\leqslant R_{p}$, we have $a_{\star}(\theta,\nu)=0$.

  2. b)

For $\nu>R_{p}$, there exists a strictly decreasing continuous function
$\theta_{c}:(R_{p},\infty)\to(0,\infty)$ such that

$a_{\star}(\theta,\nu)\quad\begin{cases}=0&\text{ for
}\theta\leqslant\theta_{c}(\nu),\\\ >0&\text{ for
}\theta>\theta_{c}(\nu).\end{cases}$

  3. c)

The function $\theta_{c}$ is bounded by

$\theta_{c}(\nu)\leqslant\min\left\\{\frac{p+1}{2}\nu^{-(p-1)/2}\cdot\xi_{p}(0),\frac{C_{d}\nu}{\nu-
R_{p}}\right\\}$

and satisfies,

$\displaystyle\lim_{\nu\uparrow\infty}\theta_{c}(\nu)\cdot\frac{2}{p+1}\nu^{(p-1)/2}$
$\displaystyle=\xi_{p}(0)\text{ and }\lim_{\nu\downarrow
R_{p}}\theta_{c}(\nu)\cdot(\nu-R_{p})=C_{d}R_{p}.$

  4. d)

Moreover, we have that for all $\nu>R_{p}$,

$\liminf_{\theta\downarrow\theta_{c}}a_{\star}(\theta,\nu)>0.$

######  Remark 2.5.

We expect $\mathscr{M}$ to be singleton when $a_{\star}>0$. However, to prove
that, we need detailed behavior of the $I$ function, especially close to
$R_{p}$. We currently lack this.

See Figure 2 for a pictorial description of the phase diagram.

I IIa IIb IIIa IIIb

Figure 2. Top: Phase Diagrams for $\\{R>0,I^{\prime}(R+)=0\\}$ and $\\{R=0$ or
$R>0,I^{\prime}(R+)<0\\}$, respectively. Bottom: Representative plots of
$a\mapsto W(\theta(1-a))/\theta+I(a\nu)/\nu$ function for different regions of
the phase diagram.

######  Remark 2.6.

With the scaling $\theta\nu^{-(p-1)/2}=C$ and $\nu\to\infty$, we recover the
regime considered in [CK12], the critical value being $C=\xi_{p}(0)$. This is
due to the fact that as $\nu$ becomes large, the nonlinear part of the
Hamiltonian dominates and the lattice sites decouple.

###  2.3. Typical Dispersive Function

We now take $(\theta,\nu)\in\mathscr{D}$. In practice, for fixed $\theta$ we
will have to take $\nu$ appropriately small (but not vanishing). We introduce
the prototypical object to which we will compare a typical function in the
dispersive phase.

######  Definition 2.7.

Let $\\{\zeta_{\boldsymbol{k}}\\}_{\boldsymbol{k}\in[n]^{d}}$ be i.i.d.
standard complex Gaussian random variables. Let
$\\{\phi^{\boldsymbol{k}}\\}_{\boldsymbol{k}\in[n]^{d}}$ and
$\\{\lambda_{\boldsymbol{k}}\\}_{\boldsymbol{k}\in[n]^{d}}$ respectively be
the eigenfunctions (23) and eigenvalues (22) of $-\Delta$, as defined on
$\ell^{2}(\mathds{T}^{d}_{n})$. We define the massive Gaussian free field
(MGFF) with mass parameter $y>0$ as

$\displaystyle\Psi^{y}:=\sum_{\boldsymbol{k}\in[n]^{d}}\frac{\zeta_{\boldsymbol{k}}}{\sqrt{y+\lambda_{\boldsymbol{k}}}}\phi^{\boldsymbol{k}}.$
(19)

The properties of the massive free field are well understood. We will recap
some of them in Section 4.

######  Theorem 2.8.

Let $\theta<C_{d}$ be fixed, $p<5+4\sqrt{2}$ and $\nu$ be sufficiently small.
Let $y_{N}$ be such that
$\operatorname{\mathds{E}}\left\|\Psi^{y_{N}}\right\|_{2}^{2}=N\theta$. Let
$\mathcal{A}_{N}\subset\mathds{C}^{N}$ be such that
$\mathds{P}(\Psi^{y_{N}}\in\mathcal{A}_{N})\leqslant N^{-\alpha}$ where
$\alpha>1/2$. Then we have an $\epsilon>0$ such that

$\mu_{N}(\mathcal{A}_{N})\leqslant N^{-\epsilon}.$

In the dispersive phase, any event sufficiently rare for the massive GFF with
appropriate parameter $\theta$ is also rare for the measure $\mu_{N}$.

######  Remark 2.9.

We anticipate that the upper bound on $p$ should be removable; it is a
consequence of using a particular auxiliary function to aid our proofs.

######  Corollary 2.10.

For $\Psi$ sampled from $\mu_{N}$, we have with probability approaching $1$

$\left\|\Psi\right\|_{\infty}\leqslant\sqrt{3C_{d}\log N}.$

The penalty factor of $N^{1/2}$ does not arise from the exponential tilting
due to the nonlinearity, but rather the severe restriction placed on the
allowed value of the mass, as will be discussed in Section 8.

##  3\. Background, Literature Review and Heuristics

The purpose of this section is threefold; provide a very brief survey of some
of the important characteristics of the continuum NLS, discuss the obstacles
that arise when studying them and finally describe how the discrete NLS
captures some of these phenomena without the obstacles. Thus the analysis of
the discrete case sheds some light on the continuum.

###  3.1. Criticality

In the continuum focusing NLS, the value of the nonlinearity parameter $p$ can
lead to drastically different long-term behavior and also impose different
requirements on the regularity and size of the initial data to obtain a well-
posed solution. Fundamentally, the issue is that $|\psi|^{p-1}\psi$ need not
be in $L^{2}$, and thus can lead to the development of point singularities.
The Gagliardo-Nirenberg-Sobolev (GNS) inequality allows us to use the $H^{1}$
norm to bound the $L^{p}$ norm, there is a constant depending on $p$ and $d$
such that

$\displaystyle\left\|\psi\right\|_{p}\leqslant
C_{p,d}\left\|\nabla\psi\right\|_{2}^{(p-1)d/2}\cdot\left\|\psi\right\|_{2}^{2+(2-d)(p-1)/2}.$
(20)

The regime $p<1+4/d$ is referred to as mass subcritical, $H^{1}$ initial data
is adequate for global well-posedness. The regime $p=1+4/d$ is referred to as
mass critical. In this case, $H^{1}$ data leads to a globally well-posed
solution so long as $L^{2}$ norm is smaller than a threshold depending on $p$
and $d$. When $p>1+4/d$, referred to as the mass supercritical regime, we
require an upper threshold for both $L^{2}$ and $H^{1}$ norms.

###  3.2. Soliton Solutions

The competition between the dispersion the focusing effect of the nonlinearity
yield spatially stationary solutions called solitons. Solton solutions may be
realized in one of two ways; either using the separation of variables or
through a variational characterization by minimizing the Hamiltonian subject
to a mass constraint. If $\varphi({\boldsymbol{x}})$ denotes a soliton
solution, it satisfies the following nonlinear elliptic problem for $\omega<0$

$\omega\varphi+\Delta\varphi+|\varphi|^{p-1}\varphi=0.$

Solitons are strongly localized; it can be proved that they decay
exponentially about a center. Further, the variational characterization
implies that they are radially symmetric about a center and smooth for energy
subcritical nonlinearity. Under their construction, optimizing the difference
of $L^{p+1}$ and $H^{1}$ norms, they happen to be the functions for which the
GNS inequality is realized, and the constant is sharp [WEI82].

###  3.3. Use of Invariant Measures

The invariant measure approach to studying the continuum NLS is well-known and
celebrated. The hope is that an invariant measure sheds light on the ‘typical’
behavior of the NLS. For instance, this can include questions of well-
posedness, as well as questions of types of solutions. Consider the famous
_soliton resolution conjecture_ , which states that for generic initial data,
we see a potion of mass coalescing into a soliton and a portion dispersing
away. Invariant measures are a natural means for talking about what
constitutes generic initial data.

In terms of construction, the idea is to use the intuition provided by finite-
dimensional Hamiltonian systems to obtain candidate invariant measures on
appropriate function spaces. Of course, there is no version of a Lebesgue
measure on a function space to which Liouville’s theorem can be applied.
However, as is well known in probability, we may rigorously make sense of
Gaussian measures which have ‘density’ proportional to
$\exp(-\left\|\psi\right\|_{H^{1}})$ on appropriate Wiener spaces.

For instance, in dimension one with Dirichlet boundary conditions, this is the
Brownian bridge on the torus $\mathds{T}^{d}$ with harmonic function fixed;
this corresponds to the Gaussian Free Field. The regularity of these as
distributions is well understood, and the hope is that by exponentially
tilting these Gaussian measures by $\left\|\psi\right\|_{p+1}^{p+1}$ and
employing a mass cut-off, we may obtain an invariant measure for the dynamics.
This requires verifying that the tilt is integrable with respect to the
reference Gaussian measure and that the NLS flow is defined on the support of
the measure. This technique was used to construct a candidate invariant
measure for the 1-d periodic focusing NLS by Lebowitz, Rose, and Speer
[LRS88]. They worked in the subcritical mass regime, where GNS inequality can
be applied to control the nonlinearity. Later, McKean and Vaninsky [MV94,
MV97a, MV97b] proved that this measure is indeed invariant for the flow.
Bourgain [B94] used a version of this invariant measure to prove global well-
posedness for the periodic equation. Brydges and Slade [BS96] followed a
similar approach for the two-dimensional periodic equation, with a slightly
different ultraviolet cutoff. They established a normalizable measure for mass
below a critical threshold. However, their measure is not invariant for the
flow of the NLS. Indeed, this approach breaks down in $d\geqslant 3$, where
this approach fails to yield even a normalizable measure, and the associated
Gaussian field is too rough. It is at this juncture that discretization comes
into play.

###  3.4. The Discrete NLS

We introduced the DNLS in (4), and observed that it retains the Hamiltonian
structure, akin to its continuum counterpart. The discrete setting is harder
to work with in many ways as several of the symmetries of the continuum NLS
are lost, such as Galilean invariance and rotational invariance. On the other
hand, we do not have the same regularity issues; the DNLS is globally well-
posed for $\ell^{2}$ initial data. Like the continuum equation, the focusing
DNLS admits soliton solutions. Like the continuum, they can be realized either
through the separation of variables or as minimizers of the Hamiltonian. We
discuss them in detail in Section 5.

In [WEI99], Weinstein studied the discrete focusing NLS on $\mathds{Z}^{d}$
and showed that soliton solutions of arbitrary mass could be realized for
mass-subcritical nonlinearity. On the other hand, in the mass-supercritical
case, there is a constant depending only on the lattice coupling strength and
nonlinearity parameter, denoted by $R_{p}$, such that soliton solutions can
only be realized when the mass is more than $R_{p}$; a phenomenon strikingly
familiar to the blow-up in the continuum. The precise statement is provided
later; see Lemma 5.1. This analogy is strengthened by the observation that
there is a correspondence between solutions of a large mass and solutions on
the lattice with low coupling strength; soliton solutions are increasingly
concentrated onto a single lattice site as the mass increases.

In [CK12], Chatterjee and Kirkpatrick examined the behavior of the discrete
focusing cubic (with $p=3$) NLS defined on the torus of dimension $d\geqslant
3$, via the analysis of a Gibbs measure of the form (6). Their regime of
scaling is chosen to correspond to taking a limit to the continuum; the blow-
up phenomenon is realized as a phase transition. They show that a single
parameter $\theta=\theta(\beta,B)$ governs the phase behavior. When
$\theta\geqslant\theta_{c}$, a single site acquires a positive fraction of the
mass. This is explained by the scaling regime considered; the $H^{1}$ norm
part of the Hamiltonian becomes irrelevant, and the measure may be regarded as
an exponential tilt of the uniform measure on the ball via the $\ell^{4}$
norm. The immediate conclusion is that the favored states are those where all
the mass is localized to a single site.

Discrete invariant measures were used to rigorously establish a version of the
soliton resolution conjecture by Chatterjee in [C14]. Chatterjee worked with a
microcanonical ensemble, _i.e.,_ the uniform measure defined on an
$\varepsilon-$thickening of a $2N-2$ dimensional surface defined by taking
constant values of mass and energy and showed that a function uniformly drawn
from this measure, modulo translation and phase rotation, converges in a
suitable sense to a continuum soliton of the same mass.

###  3.5. Scaling commentary

There is a natural scale invariance associated with the continuum NLS. If
$\psi(t,x)$ is a solution of (1), then for any $\lambda>0$,
$\lambda^{\frac{2}{p-1}}\psi(\lambda^{2}t,\lambda x)$ is also a solution.
Since the lattice cannot be scaled, the discrete equation does not admit any
symmetry with respect to scaling. However, we do have the following
equivalence.

######  Lemma 3.1.

Let $\psi_{x}(t)$ denote a solution of the discrete NLS on either the lattice
$\mathds{Z}^{d}$ or the discrete torus $\mathds{T}^{d}$ with lattice spacing
$h$. Then $\lambda^{\frac{2}{p-1}}\psi_{x}(\lambda^{2}t)$ is a solution to the
discrete NLS corresponding to the same graph with lattice spacing $h/\lambda$.

###### Proof.

Proof follows immediately by noting that for
$h^{\prime}=h/\lambda,\psi^{\prime}_{x}(t):=\lambda^{\frac{2}{p-1}}\psi_{x}(\lambda^{2}t)$
we have
$H_{h}(\psi)=\lambda^{d-2-\frac{4}{p-1}}H_{h{{}^{\prime}}}(\psi{{}^{\prime}})$,
and using the discrete analogue in (4). $\blacksquare$

This yields a family of equivalent ODEs on the lattice, where solutions of one
can be scaled into solutions of the other. For the discrete equation, this is
the reason why solutions with low values of lattice coupling can be placed in
correspondence with solutions of significant mass, as seen in [WEI99]. In
particular, in the model (8) if we replace $\psi$ by $\lambda\psi$ we have
equivalence between (6) and (8) as long as the parameters satisfy

$\displaystyle\beta h^{d-2}=\theta\lambda^{2},\quad\beta
h^{d}=\theta\lambda^{2}\cdot(\nu\lambda^{2}/N)^{(p-1)/2}\text{ and
}Bh^{-d}=N\lambda^{-2}.$

Solving we get the relations (10) with $\lambda=\sqrt{Nh^{d}/B}$.

To use physics terminology, this regime of scaling corresponds to taking the
infrared limit, without removing the ultraviolet cutoff, essentially allowing
$\mathds{T}^{d}_{n}$ to grow to $\mathds{Z}^{d}$. As a consequence of this
scaling, we may consider the behavior of concentrated and dispersed parts of
typical functions separately. Take a function $\psi$ in
$\ell^{2}(\mathds{T}^{d}_{n})$ with mass bounded above by $N$. We may break it
into a region where the values are of order $\sqrt{N}$ and a region where they
are of strictly lower order. Let the region of concentration be $U$. The
energy of the restriction $\psi_{U}$ is then expressible in terms of the
$\mathds{Z}^{d}$ Hamiltonian as

$\mathcal{H}_{N}(\psi_{U})\approx\frac{N}{\nu}\mathcal{H}\left(\sqrt{\frac{\nu}{N}}\psi_{U}\right)$

where $\sqrt{\nu/N}\cdot\psi_{U}$ scales to yield a valid function in
$\ell^{2}(\mathds{Z}^{d})$.

As for the dispersive part, whenever the order of typical values of $\psi_{x}$
is lower than $N$, the nonlinearity does not contribute, and we see Gaussian
Free Field behavior for $\psi_{U^{c}}$. Moreover, the contribution of this
portion to the free energy is non-trivial. The analysis of this regime is what
makes this article novel; usually the scaling is chosen such that a function
sampled from the measure converges to a soliton, our work strongly suggests
(yet falls short of explicitly characterizing) the behavior of the
fluctuations about this soliton, in the vein of studying fluctuations for
various random surface models. Indeed, this is the case for our analysis of
the typical function in the dispersive phase. The fact that there is no
soliton essentially corresponds to the fact that no centering is required; we
only see the fluctuations.

###  3.6. Notations

The following are fixed for the entire article

  1. i.

We will denote the integer lattice by $\mathds{Z}^{d}$ and the discrete torus
of side length $n$ by $\mathds{T}^{d}_{n}$.

  2. ii.

Vertices in $\mathds{T}^{d}_{n}$ or $\mathds{Z}^{d}$ will be denoted by
${\boldsymbol{x}}$.

  3. iii.

The dual variable to ${\boldsymbol{x}}$ in the sense of the Fourier transform
defined on $\mathds{T}_{n}^{d}$ will be denoted by $\boldsymbol{k}$.

  4. iv.

$N$ will always denote $n^{d}$.

  5. v.

$\mathds{C}^{N}$, as is standard will denote the complex vector space of
dimension $N$, with the standard inner product.

  6. vi.

$\theta$ and $\nu$ will be positive real numbers denoting the inverse
temperature and the coupling constant, respectively.

  7. vii.

$\mathscr{S}$ and $\mathscr{D}$ are subsets of $[0,\infty)^{2}$ denoting the
solitonic and dispersive regions of the parametric plane $(\theta,\nu)$.

  8. viii.

$\mathscr{M}(\theta,\nu)$ is the collection of optimizers of the variational
formula defining the free energy.

As best as possible, we work with the following conventions. We also highlight
frequently recurring examples.

  1. i.

Subsets of $\mathds{Z}^{d}$ or $\mathds{T}^{d}$ will be denoted by capitalized
Roman letters such as $U$ or $V$.

  2. ii.

Subsets of the function spaces $\ell^{2}(\mathds{T}^{d})$ will be denoted by
calligraphic letters such as $\mathcal{A}$ or $\mathcal{B}$.

  3. iii.

Functions in $\ell^{2}(\mathds{Z}^{d})$ or $\ell^{2}(\mathds{T}^{d})$ will be
denoted by small Greek letters such as $\psi$ or $\phi$, with the following
recurring examples.

    1. (a)

Restrictions of a function $\psi$ to a subset $U$ will be denoted as
$\psi_{U}$.

    2. (b)

Discrete solitons of mass $a$ will be denoted as $\varphi^{a}$.

    3. (c)

Eigenfunctions of the $\mathds{T}^{d}_{n}$ Laplacian will be denoted as
$\phi^{\boldsymbol{k}}$ with $\boldsymbol{k}\in[n]^{d}$.

    4. (d)

Functions in the orthogonal complement of the subspace spanned by
$\\{\phi^{\boldsymbol{0}}\\}$ will be denoted by $\psi^{\perp}$.

  4. iv.

Random fields will be denoted by capital Greek letters such as $\Psi$ and
$\Phi$, with the following recurring examples,

    1. (a)

$\Psi^{U,y}$ will denote the massive Dirichlet Gaussian Free Field taking
values in $\ell^{2}(U)$.

    2. (b)

$\Psi^{\boldsymbol{0},y}$ will denote the massive zero average Gaussian Free
Field taking values in $\ell^{2}(\mathds{T}^{d}_{n})$.

    3. (c)

The superscript $y$ will be dropped when we have $y=0$, when appropriate.

  5. v.

The Laplacians under consideration will be variations of $\Delta$.

    1. (a)

Restriction to the complement of the kernel will be denoted as
$\Delta^{\perp}$.

    2. (b)

Dirichlet Laplacians on $U$ will be denoted as $\Delta^{0}_{U}$.

  6. vi.

Constants will usually be denoted by variations on the letter $C$. The most
frequently recurring standard constant is the following.

    1. (a)

$C_{d}$ will denote the limiting mass per site of $\Psi^{\boldsymbol{0}}$.

Important exceptions to the conventions listed above are unavoidable for
various reasons, such as consistency with prior literature. We list them
below.

  1. i.

The Hamiltonians will always be denoted with variations of $\mathcal{H}$.
There are three cases

    1. (a)

$\mathcal{H}_{c}$ is the continuum Hamiltonian and will not be refrerred to
beyond the background material.

    2. (b)

$\mathcal{H}$ is the discrete Hamiltonian for the DNLS defined on
$\mathds{Z}^{d}$, with $h=1$. See (5).

    3. (c)

$\mathcal{H}_{N}$ is our scale-dependent model Hamiltonian with which we
define the Gibbs measure of interest.

  2. ii.

The following capital Roman Letters have specific meanings, and are not
subsets of lattice sites.

    1. (a)

The function $I(a)$ for $a\geqslant 0$ will always denote the minimum
$\mathcal{H}$ for fixed mass $a$, $\mathcal{H}$ as defined in (5). See (11).

    2. (b)

The function $K(y)$ will always denote the limiting scaled log determinant of
$y-\Delta^{\perp}$. See (12).

    3. (c)

The function $W(b)$ for $b>0$ will always denote the Legendre transform of
$K$. See (13).

    4. (d)

The function $L$ will always denote the inverse of $K^{\prime}$. See (27)

    5. (e)

$R_{p}$ will always denote the mass threshold for soliton formation. See Lemma
5.1

  3. iii.

Parameters $\theta>0$ and $\nu>0$ are the inverse temperature and coupling
constant for the nonlinearity in (9), and are not functions in $\ell^{2}$.

Along the way, we will define certain auxiliary functions and random variables
to make calculations more convenient to express. These will be defined in a
context-appropriate fashion.

###  3.7. Organization of the Paper

The article is organized as follows. In Section 4, we provide a description of
the Discrete Gaussian Free Field and explain its importance for our analysis.
We then prove the convergence of its limiting free energy, and of that
conditioned to have a specified mass. We also establish some useful
$\ell^{\infty}$ bounds. In Section 5, we discuss soliton solutions of the
DNLS. In particular, we construct exponentially decaying minimizers for (5).
We also establish properties of the function $I$. In Section 6, we combine
insights from Sections 4 and 5 to prove the convergence of the limiting free
energy, that is Theorem 2.3. In Section 7, we analyze the phases. In
particular, we demonstrate that there are two regimes of optimal mass
allocation, one where we have a non-trivial soliton, and one where we do not.
That is, we prove Theorem 2.4. In Section 8, we provide commentary on the
behavior of a typical function in the dispersive phase. We provide an explicit
comparison between the reference measure corresponding to the linear part of
the Hamiltonian and the massive Gaussian Free Field. We then show that the
tilt corresponding to the nonlinearity is integrable with respect to the
massive free field. A combination of these two results verifies Theorem 2.8.
As an immediate corollary, we have that that a typical function in the
dispersive phase is bounded above in probability by $\sqrt{3C_{d}\log N}$. We
conclude the article with some interesting questions for the future.

##  4\. Gaussian Free Field

The dispersive contribution to the free energy is given by the integral

$\displaystyle M_{N}(b,\varepsilon):=\int_{\left\|\psi\right\|^{2}_{2}\in
N(b-\varepsilon,b+\varepsilon)}\exp\bigl{(}-\left\|\nabla\psi\right\|_{2}^{2}\bigr{)}\,\mathbf{d}\psi$
(21)

where $b\geqslant 0,\varepsilon>0$. Understanding the asymptotics as
$N\to\infty$ is thus of fundamental importance to this article. This section
is first and foremost dedicated to establishing the following theorem.

######  Theorem 4.1.

Let $b>0$, and $\varepsilon>0$ be a positive number such that
$N\varepsilon^{d}\gg 1$. We have

$\displaystyle\left|\frac{1}{N}\log
M_{N}(b,\varepsilon)-\pi+W(b)\right|\leqslant 2\varepsilon
L(b)+2/(N^{2}\varepsilon^{2}\cdot
m_{N}(2))+\frac{1}{N}\log((b+\varepsilon)N).$

where $m_{N}(\cdot)$ is as per Lemma 4.4.

What prevents the immediate representation of the integral as an expectation
with respect to a Gaussian random variable is the fact that the quadratic form
in the exponential is degenerate; it corresponds to $-\Delta$ which has a non-
trivial kernel. However, we can still relate this integral to the large
deviations of the mass of an appropriate Gaussian field called the
zero–average Gaussian Free Field (GFF). We will define the GFF and evaluate
the asymptotic via a combination of two probabilistic techniques, exponential
tilting, and concentration. We will then conclude the section with some
maximum estimates will be of importance later.

###  4.1. Definitions

The Laplacian $\Delta$ is a translation-invariant operator on $\mathds{C}^{V}$
with respect to the standard basis and is thus diagonalized by the Fourier
basis. It is well-known that the eigenvalues of the Laplacian on the discrete
torus with vertex set $[n]^{d}$ are given by

$\displaystyle\lambda_{\boldsymbol{k}}=4\sum_{i=1}^{d}\sin^{2}(\pi
k_{i}/n)=f(\boldsymbol{k}/n)\text{ for }\boldsymbol{k}\in[n]^{d}.$ (22)

The corresponding eigenfunctions are

$\displaystyle\phi^{\boldsymbol{k}}_{\boldsymbol{x}}=\frac{1}{\sqrt{N}}\exp\left(2\pi{\mathrm{i}\mkern
2.0mu}\cdot{(\boldsymbol{k}\cdot{\boldsymbol{x}})}/n\right),\qquad{\boldsymbol{x}},\boldsymbol{k}\in[n]^{d}.$
(23)

######  Definition 4.2 (Massive zero–average GFF).

Let $y\geqslant 0$. We define the massive zero–average free field
$\Psi^{\boldsymbol{0},y}\in\mathds{C}^{V}$ as the random vector

$\Psi^{\boldsymbol{0},y}:=\sum_{\boldsymbol{k}\neq\boldsymbol{0}}\frac{\zeta_{\boldsymbol{k}}}{\sqrt{\lambda_{\boldsymbol{k}}+y}}\phi^{\boldsymbol{k}},$

where $\lambda_{\boldsymbol{k}}$ are the eigenvalues of $-\Delta$ as in (22),
$\phi^{\boldsymbol{k}}$ are the corresponding eigenfunctions as in (23) and
$\zeta_{\boldsymbol{k}}$ are i.i.d. standard complex Gaussian random
variables.

We may explicitly write down the density of the zero–average free field, which
is expressible as a Gibbs measure in its own right. We will be working with
the subspace $\mathds{C}^{N}/\\{\phi^{\boldsymbol{0}}\\}$, that is the
subspace of all $\psi^{\perp}$ that are orthogonal to $\phi^{\boldsymbol{0}}$.
Note, the restriction is negative definite and is therefore invertible. We
have

$\mathds{P}(\Psi^{\boldsymbol{0},y}\in\mathcal{A}):=\frac{1}{Z^{\boldsymbol{0},y}}\int_{\mathcal{A}}\exp\left(-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}-y\left\|\psi^{\perp}\right\|_{2}^{2}\right)\mathbf{d}\psi^{\perp}.$

where $\mathbf{d}\psi^{\perp}$ denotes the volume element on
$\mathds{C}/\\{\phi^{\boldsymbol{0}}\\}$. We will denote the restriction of
the Laplacian on this space as $\Delta^{\perp}$. The partition function is
given by

$Z^{\boldsymbol{0},y}:=\pi^{N-1}/\det(y-\Delta^{\perp}).$

We refer the interested reader to [S07] and [A19] for more details on GFF and
zero–average GFF on the discrete torus, respectively.

######  Remark 4.3.

The operator $y-\Delta$ for $y>0$ is positive definite, therefore we may
define a Gaussian process with covariance $(y-\Delta)^{-1}$ without the
restriction to $\mathds{C}^{N}/\mathrm{span}\\{\phi^{\boldsymbol{0}}\\}$. This
is exactly the massive Gaussian Free Field, seen in (19).

###  4.2. Analysis of the Limiting Free Energy

From here onwards, the graph under consideration will be the entirety of the
discrete torus $\mathds{T}^{d}$, and we will describe the properties of the
measure associated with the field $\Psi^{\boldsymbol{0},y}$. The associated
mean free energy is given by

$\displaystyle\frac{1}{N}\log
Z^{\boldsymbol{0},y}=\frac{N-1}{N}\log\pi-\frac{1}{N}\sum_{\boldsymbol{k}\neq\boldsymbol{0}}\log(\lambda_{\boldsymbol{k}}+y).$

In order to more compactly express our results, for
${\boldsymbol{x}}\in[0,1]^{d}$, we define

$\displaystyle f({\boldsymbol{x}}):=4\sum_{i=1}^{d}\sin^{2}(\pi
x_{i}),\quad{\boldsymbol{x}}=(x_{1},x_{2},\ldots,x_{d})\in[0,1]^{d}$ (24)

and

$\displaystyle
g_{y}({\boldsymbol{x}}):=\log\left(y+f({\boldsymbol{x}})\right).$

Let ${\boldsymbol{x}}$ denote a uniform random variable on $[0,1]^{d}$. We
define ${\boldsymbol{x}}_{n}:=\lfloor
n{\boldsymbol{x}}\rfloor/n\sim\text{Uniform}(\\{0,1/n,\ldots,1-1/n\\}^{d})$
and

$\displaystyle
K_{N}(y):=\frac{1}{N}\sum_{\boldsymbol{k}\neq\boldsymbol{0}}\log(y+\lambda_{\boldsymbol{k}})=\operatorname{\mathds{E}}g_{y}({\boldsymbol{x}}_{n})\mathds{1}_{{\boldsymbol{x}}_{n}\neq\boldsymbol{0}}.$

Observe that the expected mass of $\Psi^{\boldsymbol{0},y}$ is given by

$\operatorname{\mathds{E}}\left\|\Psi^{\boldsymbol{0},y}\right\|_{2}^{2}=-\frac{d}{dy}\log
Z^{\boldsymbol{0},y}=N\cdot K_{N}^{\prime}(y).$

Recall the function $K$ introduced in (12). By definition,
$K=\lim_{n\to\infty}K_{N}$. Clearly we should have

$\displaystyle
K(y)=\int_{[0,1]^{d}}g_{y}({\boldsymbol{x}})\,d{\boldsymbol{x}}.$ (25)

We will prove this convergence now, and as an abuse of notation, we use (25)
as the definition of $K$. We refer to the following lemma from [DK21], which
is important for establishing rates of convergence and follows quite easily.

######  Lemma 4.4 ([DK21]*Lemma $2.1$).

Let $\\{\lambda_{\boldsymbol{k}}\\}_{\boldsymbol{k}\neq 0}$ be the eigenvalues
of $-\Delta^{\perp}$. Then we have for any $p>0$,

$m_{N}(p):=\frac{1}{N}\sum_{\boldsymbol{k}\neq\boldsymbol{0}}\lambda_{\boldsymbol{k}}^{-p}\simeq\begin{cases}1&\text{
if }d>2p\\\ \log N&\text{ if }d=2p\\\ N^{-1+2p/d}&\text{
otherwise}.\end{cases}$

This lemma is first of use in order to calculate the rate of convergence of
the scaled expected mass, i.e.
$\operatorname{\mathds{E}}\left\|\Psi^{0,y}\right\|_{2}^{2}$.

######  Lemma 4.5.

For $y\geqslant 0$, we have a constant $c$, depending only on $d\geqslant 3$,
such that

$\displaystyle|K_{N}(y)-K(y)|\leqslant c\cdot N^{-1/d}\text{ and
}|K^{\prime}_{N}(y)-K^{\prime}(y)|\leqslant c\cdot
N^{-1/d}\cdot(1+\mathbf{1}_{d=3}\cdot\log N).$

###### Proof.

Let

$g_{y}({\boldsymbol{x}}):=\log(y+f({\boldsymbol{x}}))\text{ and
}g^{\prime}_{y}({\boldsymbol{x}}):=\partial_{y}g_{y}({\boldsymbol{x}})=(y+f({\boldsymbol{x}}))^{-1},\quad{\boldsymbol{x}}\in[0,1]^{d}.$

It is clear that

$\displaystyle K_{N}(y)$
$\displaystyle=\operatorname{\mathds{E}}g_{y}({\boldsymbol{x}}_{n})\mathds{1}_{{\boldsymbol{x}}_{n}\neq\boldsymbol{0}},\text{
}K(y)=\operatorname{\mathds{E}}g_{y}({\boldsymbol{x}}),$ $\displaystyle
K^{\prime}_{N}(y)$
$\displaystyle=\operatorname{\mathds{E}}g^{\prime}_{y}({\boldsymbol{x}}_{n})\mathds{1}_{{\boldsymbol{x}}_{n}\neq\boldsymbol{0}}\text{
and }K^{\prime}(y)=\operatorname{\mathds{E}}g^{\prime}_{y}({\boldsymbol{x}}).$

Thus, we have, for any $y\geqslant 0$, we have

$\displaystyle\left|K(y)-K_{N}(y)\right|$
$\displaystyle\leqslant\operatorname{\mathds{E}}|g_{y}({\boldsymbol{x}})|\mathds{1}_{{\boldsymbol{x}}_{n}=\boldsymbol{0}}+\operatorname{\mathds{E}}\left|g_{y}({\boldsymbol{x}})-g_{y}({\boldsymbol{x}}_{n})\right|\mathds{1}_{{\boldsymbol{x}}_{n}\neq\boldsymbol{0}}$
$\displaystyle\text{ and }\left|K^{\prime}(y)-K^{\prime}_{N}(y)\right|$
$\displaystyle\leqslant\operatorname{\mathds{E}}g^{\prime}_{y}({\boldsymbol{x}})\mathds{1}_{{\boldsymbol{x}}_{n}=\boldsymbol{0}}+\operatorname{\mathds{E}}\left|g^{\prime}_{y}({\boldsymbol{x}})-g^{\prime}_{y}({\boldsymbol{x}}_{n})\right|\mathds{1}_{{\boldsymbol{x}}_{n}\neq\boldsymbol{0}}.$

It is easy to chek that,

$\operatorname{\mathds{E}}|g_{y}({\boldsymbol{x}})|\mathds{1}_{{\boldsymbol{x}}_{n}=\boldsymbol{0}}\leqslant\operatorname{\mathds{E}}|\log(y+f({\boldsymbol{x}})|\mathds{1}_{{\boldsymbol{x}}_{n}=\boldsymbol{0}}\leqslant(1+|\log
y|)\cdot N^{-1}$

and

$\operatorname{\mathds{E}}g^{\prime}_{y}({\boldsymbol{x}})\mathds{1}_{{\boldsymbol{x}}_{n}=\boldsymbol{0}}\leqslant\operatorname{\mathds{E}}f({\boldsymbol{x}})^{-1}\mathds{1}_{{\boldsymbol{x}}_{n}=\boldsymbol{0}}\leqslant
cN^{2/d-1}\leqslant cN^{-1/d}.$

Note that with $y>0$ fixed, $g_{y}(\cdot)$ is smooth, with bounded derivatives
of all orders. Moreover, when $\left\|{\boldsymbol{x}}_{n}\right\|>0$, we can
bound

$\displaystyle\left|g_{y}({\boldsymbol{x}})-g_{y}({\boldsymbol{x}}_{n})\right|$
$\displaystyle\leqslant\left\|{\boldsymbol{x}}-{\boldsymbol{x}}_{n}\right\|\cdot\left\|\nabla
g_{y}({\boldsymbol{x}}_{n}^{*})\right\|\leqslant cN^{-1/d}\cdot
f({\boldsymbol{x}}_{n})^{-1/2}$ $\displaystyle\text{ and
}\left|g^{\prime}_{y}({\boldsymbol{x}})-g^{\prime}_{y}({\boldsymbol{x}}_{n})\right|$
$\displaystyle\leqslant\left\|{\boldsymbol{x}}-{\boldsymbol{x}}_{n}\right\|\cdot\left\|\nabla
g^{\prime}_{y}({\boldsymbol{x}}_{n}^{*})\right\|\leqslant cN^{-1/d}\cdot
f({\boldsymbol{x}}_{n})^{-3/2}.$

Here we used the fact that

$\left\|\nabla f({\boldsymbol{x}})\right\|^{2}=\sum_{i=1}^{d}(8\sin(\pi
x_{i})\cos(\pi x_{i}))^{2}\leqslant 16f({\boldsymbol{x}}).$

Summing, we get that

$\left|K(y)-K_{n}(y)\right|\leqslant cN^{-1/d}m_{N}(1/2)\text{ and
}\left|K^{\prime}(y)-K^{\prime}_{N}(y)\right|\leqslant cN^{-1/d}m_{N}(3/2),$

where $m_{n}$ is as given in Lemma 4.4. This completes the proof.
$\blacksquare$

We now take the opportunity to introduce an important dimension dependent
constant. We define

$\displaystyle C_{d}:=K^{\prime}(0),$ (26)

which is clearly finite for $d\geqslant 3$.

######  Remark 4.6.

The constant $C_{d}$ has an important interpretation in probability, it is the
expected number of returns to ${\boldsymbol{x}}$ for a simple symmetric random
walk started at ${\boldsymbol{x}}$ in $\mathds{Z}^{d}$. Clearly,
$C_{d}<\infty$ when $d\geqslant 3$ and is infinite otherwise due to the
recurrence of the random walk.

It is easy to check that $2dC_{d}\geqslant
2d/\operatorname{\mathds{E}}(f({\boldsymbol{x}}))=1$ for all $d\geqslant 3$
and converges to $1$ as $d\to\infty$. In Table 1 we provide the numerical
values of $C_{d}$ for $d=3,4,\cdots,10$.

$d$ | 3 | 4 | 5 | 6 | 6 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---
$C_{d}$ | 0.252 | 0.155 | 0.116 | 0.093 | 0.078 | 0.067 | 0.059 | 0.053
Table 1. Numerical values for $C_{d}$

Since $K^{\prime}$ is decreasing and convex, we may define an inverse function

$\displaystyle L:(0,C_{d}]\to[0,\infty)$ (27)

which is also decreasing, convex and by hypothesis satisfies
$K^{\prime}(L(b))=b$. We extend $L$ by defining $L(b)=0$ for $b>C_{d}$. Note
that, $bL(b)\leqslant 1$ for all $b>0$ and

$\displaystyle W(b)=\inf_{y\,:\,K^{\prime}(y)\leqslant
b}(K(y)-yb)=\begin{cases}K(L(b))-bL(b)&\text{ if }b<C_{d}\\\ K(0)&\text{ if
}b\geqslant C_{d}.\end{cases}$ (28)

The function $W$ is singular at $0$, however the divergence can be well
understood.

######  Lemma 4.7.

The function $W$ given in (13) is a decreasing convex function of $b$ with
$W^{\prime}(b)=-L(b)$ and $\lim_{b\to 0+}(W(b)+\log eb)=0$. Moreover,
$\widehat{W}$ defined by

$\displaystyle\widehat{W}(b):=W(b)+\log eb=\int_{0}^{b}(s^{-1}-L(s))\,ds,$
(29)

is an increasing concave function for $b\in(0,C_{d}]$.

###### Proof of Lemma 4.7.

The function $W$ is decreasing and convex follows from the fact that
$W^{\prime}(b)=-L(b)\leqslant 0$ and
$W^{\prime\prime}(b)=-L^{\prime}(b)=-1/K^{\prime}(L(b))=1/\operatorname{\mathds{E}}(L(b)+f({\boldsymbol{x}}))^{-2}>0$.

Note that

$b=K^{\prime}(L(b))=\operatorname{\mathds{E}}(L(b)+f({\boldsymbol{x}}))^{-1}\geqslant(L(b)+\operatorname{\mathds{E}}f({\boldsymbol{x}}))^{-1}=(L(b)+2d)^{-1}$

implies that $1/b-L(b)\leqslant 2d$. Moreover, with $y=L(b)>0$, we have

$\displaystyle yb=yK^{\prime}(y)$
$\displaystyle=1-\operatorname{\mathds{E}}f({\boldsymbol{x}})(y+f({\boldsymbol{x}}))^{-1}$
$\displaystyle\leqslant
1-(y\operatorname{\mathds{E}}f({\boldsymbol{x}})^{-1}+1)^{-1}=1-(C_{d}y+1)^{-1}=y(y+1/C_{d})^{-1}.$

Simplifying we get

$1/b-L(b)\geqslant 1/C_{d}.$

The last conclusion in Lemma 4.7 follows from the fact that
$\widehat{W}^{\prime}(b)=1/b-L(b)>0$ and

$\displaystyle\widehat{W}^{\prime\prime}(b)$
$\displaystyle=-b^{-2}-L^{\prime}(b)$
$\displaystyle=b^{-2}L^{\prime}(b)\cdot(-1/L^{\prime}(b)-b^{2})=b^{-2}L^{\prime}(b)\cdot(\operatorname{\mathds{E}}(L(b)+f({\boldsymbol{x}}))^{-2}-b^{2})\leqslant
0$

as
$\operatorname{\mathds{E}}(L(b)+f({\boldsymbol{x}}))^{-2}>(\operatorname{\mathds{E}}(L(b)+f({\boldsymbol{x}}))^{-1})^{2}=b^{2}$
and $L^{\prime}(b)\leqslant 0$. $\blacksquare$

Figure 3. Plot of $W$ functions for $d=3,4,\ldots,8$; Lower function
corresponds to higher $d$.

See Figure 3 for plot of the $W$ function in dimensions $d=3,4,\ldots,10$.

###  4.3. Concentration of Mass

We now return to our integral of interest,

$M_{N}(b,\varepsilon)=\int_{\left\|\psi\right\|_{2}^{2}\in
N(b-\varepsilon,b+\varepsilon)}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi,$

and the proof of Theorem 4.1. We relate the integral to the concentration of
mass of $\Psi^{\boldsymbol{0},L(b)}$, and analyze two distinct cases depending
on whether $b\leqslant C_{d}$ or $b>C_{d}$. Let $\\{X_{\boldsymbol{k}}\\}$ be
i.i.d. exponential with rate 1 random variables. A simple change of variables
argument tells us that the mass of $\Psi^{\boldsymbol{0},y}$ may be
represented as

$\displaystyle\Gamma_{N,y}:=\left\|\Psi^{\boldsymbol{0},y}\right\|^{2}_{2}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\sum_{\boldsymbol{k}\neq\boldsymbol{0}}\frac{1}{\lambda_{\boldsymbol{k}}+y}\cdot
X_{\boldsymbol{k}}.$ (30)

######  Lemma 4.8.

Let $0<b<C_{d}$ with $C_{d}$ and $L$ from (26) and (27), respectively and
$n\varepsilon^{d}\gg 1$. Then we have, for some constant $c>0$

$\displaystyle\operatorname{\mathds{P}}\left(\Gamma_{N,L(b)}\in
N(b-\varepsilon,b+\varepsilon)\right)\geqslant 1-cm_{N}(2)/N\varepsilon^{2}.$

###### Proof.

The proof hinges on the fact that the expectation of $N^{-1}\Gamma_{N,y}$ is
$K^{\prime}_{N}(y)/N$, which converges $K^{\prime}(y)$ as $N\to\infty$.
Moreover, $K^{\prime}$ has the well defined inverse
$L(\cdot):(0,C_{d}]\to[0,\infty)$. What this means in practice is that we may
choose $y$ such that the limiting mean is $b$, so long as $b\in(0,C_{d}]$.
Using Lemma 4.5, we know that

$\displaystyle|K^{\prime}_{N}(L(b))-b|\leqslant cN^{-1/d}m_{N}(3/2).$

It is therefore adequate to verify concentration of $\Gamma_{N,y}/N$ about
$K_{N}^{\prime}(y)$. Applying Chebyshev’s inequality to (30),

$\displaystyle\operatorname{\mathds{P}}\left(\left|\Gamma_{N,y}-NK_{N}^{\prime}(y)\right|\geqslant
N\varepsilon\right)\leqslant(N\varepsilon)^{-2}\cdot\sum_{\boldsymbol{k}\neq\boldsymbol{0}}(\lambda_{\boldsymbol{k}}+y)^{-2}.$

Since $y\geqslant 0$,

$\displaystyle\operatorname{\mathds{P}}\left(\left|\Gamma_{N,y}-NK_{N}^{\prime}(y)\right|\geqslant
N\varepsilon\right)\leqslant(N\varepsilon)^{-2}\cdot\sum_{\boldsymbol{k}\neq\boldsymbol{0}}\lambda_{\boldsymbol{k}}^{-2}=m_{N}(2)/N\varepsilon^{2}.$

The last line follows by Lemma 4.4. This verifies the requisite concentration.
$\blacksquare$

######  Corollary 4.9.

Let $0<b<C_{d}$, $L(\cdot)$ as in (27), and $\phi^{\boldsymbol{0}}$ as in
(23). Let

$\mathcal{A}=\left\\{\psi^{\perp}\in\mathds{C}^{V}/\\{\phi^{\boldsymbol{0}}\\}:N(b-\varepsilon)\leqslant\left\|\psi^{\perp}\right\|_{2}^{2}\leqslant
N(b+\varepsilon)\right\\}.$

Then we have

$\left|\frac{1}{N}\log\int_{\mathcal{A}}\exp\left(-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}\right)\mathbf{d}\psi^{\perp}-\bigl{(}\pi-W(b)\bigr{)}\right|\leqslant
2\varepsilon
L(b)+\left|\log\left(1-\frac{cm_{N}(2)}{N\varepsilon^{2}}\right)\right|.$

###### Proof.

This corollary follows almost directly from verifying

$\displaystyle
e^{N(b-\varepsilon)L(b)}\left(1-\frac{cm_{N}(2)}{N\varepsilon^{2}}\right)Z^{\boldsymbol{0},L(b)}\leqslant\int_{\mathcal{A}}\exp\left(-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}\right)\mathbf{d}\psi^{\perp}\leqslant
Z^{\boldsymbol{0},L(b)}e^{NL(b)(b+\varepsilon)}.$ (31)

For efficiency of notation, for $y>0$ we will denote the quadratic form
$\left\|\nabla\psi^{\perp}\right\|_{2}^{2}+y\left\|\psi^{\perp}\right\|_{2}^{2}$
by $Q^{y}(\psi^{\perp})$. The following bounds may be immediately verified.

$e^{NL(b)(b-\varepsilon)}\int_{\mathcal{A}}\exp({-Q^{L(b)}(\psi^{\perp})})\mathbf{d}\psi\leqslant\int_{\mathcal{A}}\exp\left({-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}}\right)\mathbf{d}\psi$

and

$\int_{\mathcal{A}}\exp\left({-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}}\right)\mathbf{d}\psi\leqslant
e^{NL(b)(b+\varepsilon)}\int_{\mathcal{A}}\exp({-Q^{L(b)}(\psi^{\perp})})\mathbf{d}\psi.$

We begin with the upper bound to (31) as it is easier to establish, it follows
simply by enlarging the region of integration to all of $\mathds{C}^{V}$:

$e^{NL(b)(b+\varepsilon)}\int_{\left\|\psi^{\perp}\right\|_{2}^{2}\in
N(b-\varepsilon,b+\varepsilon)}\exp(-Q^{L(b)}(\psi^{\perp}))\mathbf{d}\psi\leqslant\pi^{N-1}\exp\bigl{(}NL(b)(b+\varepsilon)-NK_{N}(L(b))\bigr{)}.$

Immediately, we may conclude that

$\frac{1}{N}\log\int_{\left\|\psi^{\perp}\right\|_{2}^{2}\in
N(b-\varepsilon,b+\varepsilon)}\exp\left(-\left\|\nabla\psi\right\|_{2}^{2}\right)\mathbf{d}\psi\leqslant\pi+L(b)(b+\varepsilon)-K_{N}(L(b)).$

Next, we show that the lower bound converges to the same limit as the upper
bound. It is at this point that we introduce the concentration estimates. We
have

$\displaystyle\frac{1}{Z^{\boldsymbol{0},L(b)}}\int_{\left\|\psi\right\|_{2}^{2}\in
N(b-\varepsilon,b+\varepsilon)}\exp(-Q^{L(b)}(\psi))\mathbf{d}\psi=\operatorname{\mathds{P}}\biggl{(}\left\|\Psi^{\boldsymbol{0},L(b)}\right\|^{2}\in
N(b-\varepsilon,b+\varepsilon)\biggr{)}.$

Lemma 4.8 may now be directly applied. $\blacksquare$

######  Remark 4.10.

It is a trivial extension of the above corollary that

$\frac{1}{N}\log\int_{\left\|\psi^{\perp}\right\|_{2}^{2}\leqslant
N(b+\varepsilon)}\exp\left(\left\|\nabla\psi^{\perp}\right\|_{2}^{2}\right)\mathbf{d}\psi^{\perp}$

converges to $\pi-W(b)$ with the same bound for the rate of convergence. This
fact is required for the proof of Theorem 4.1, but the proof is identical to
that above and is therefore omitted.

We have now assembled all the ingredients required to prove Theorem 4.1.

###### Proof of Theorem 4.1.

Let $\psi$ be such that
$N(b-\varepsilon)\leqslant\left\|\psi\right\|_{2}^{2}\leqslant
N(b+\varepsilon)$. We orthogonally decompose $\psi$ with respect to to
$\phi^{\boldsymbol{0}}$ obtaining
$\psi=c_{\boldsymbol{0}}\phi^{\boldsymbol{0}}+\psi^{\perp}$. Thus,

$N(b-\varepsilon)\leqslant|c_{0}|^{2}+\left\|\psi^{\perp}\right\|_{2}^{2}\leqslant
N(b+\varepsilon).$

Let $b^{\prime}=\min\\{b,C_{d}\\}$. We define

$\displaystyle\mathcal{A}_{1}:=\left\\{c_{\boldsymbol{0}}\phi_{\boldsymbol{0}}:b-b^{\prime}-\frac{\varepsilon}{2}\leqslant\frac{1}{N}|c_{0}|^{2}\leqslant
b-b^{\prime}+\frac{\varepsilon}{2}\right\\}\times\left\\{\psi^{\perp}:b^{\prime}-\frac{\varepsilon}{2}\leqslant\frac{1}{N}\left\|\psi^{\perp}\right\|_{2}^{2}\leqslant
b^{\prime}+\frac{\varepsilon}{2}\right\\}.$ (32)

and

$\displaystyle\mathcal{A}_{2}:=\left\\{c_{0}\phi_{\boldsymbol{0}}:|c_{0}|^{2}\leqslant
N(b+\varepsilon)\right\\}\times\left\\{\psi^{\perp}:\left\|\psi^{\perp}\right\|_{2}^{2}\leqslant
N(b+\varepsilon)\right\\}.$ (33)

It is clear that

$\mathcal{A}_{1}\subset\mathcal{A}\subset\mathcal{A}_{2}.$

The proof is reduced to showing that the integrals of
$\exp(-\left\|\nabla\psi\right\|_{2}^{2})$ over $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ are logarithmically equivalent. The upper bound is easier,
so we establish it first. We have

$\int_{\mathcal{A}_{2}}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi=\pi
N(b+\varepsilon)\cdot\int_{\left\|\psi^{\perp}\right\|_{2}^{2}\leqslant
N(b+\varepsilon)}\exp\left(-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}\right)\mathbf{d}\psi^{\perp}$

Applying Corollary 4.9,

$\frac{1}{N}\log\int_{\mathcal{A}_{2}}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi\leqslant\pi-W(b)+\frac{1}{N}\log(b+\varepsilon)N$

Now as for the lower bound,

$\int_{\mathcal{A}_{1}}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi\geqslant\pi
N\varepsilon\cdot\int_{\left\|\psi^{\perp}\right\|_{2}^{2}\in
N(b^{\prime}-\varepsilon/2,b^{\prime}+\varepsilon/2)}\exp\left(-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}\right)\mathbf{d}\psi^{\perp}.$

The concluding step is to again use Corollary 4.9 $\blacksquare$

###  4.4. Bounds on the Maximum

In order to address how the soliton is distinguishable from the background
noise, we need upper bounds on the values that the free field can take at a
point. Additionally, maximum bounds are crucial for the stitching procedure
required in Section 6.2. The most crucial result in this section is the
following.

######  Theorem 4.11.

Let $C_{d}$ be as defined in (26) and $b\in(0,\infty)$. We have

$\displaystyle\left(\int_{\mathcal{A}^{\prime}}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi\right)$
$\displaystyle/\left(\int_{\mathcal{A}}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi\right)$
$\displaystyle\geqslant\frac{b-b^{\prime}+\varepsilon}{b+\varepsilon}\cdot
e^{-2N\varepsilon L(b)}\cdot\left(1-2cm_{N}(2)/N\varepsilon^{2}\right)$ (34)

where

$\displaystyle\mathcal{A}$
$\displaystyle=\\{\psi:N(b-\varepsilon)\leqslant\left\|\psi\right\|_{2}^{2}\leqslant
N(b+\varepsilon)\\}$ $\displaystyle\text{and }\mathcal{A}^{\prime}$
$\displaystyle=\mathcal{A}\cap\\{\psi:\left\|\psi\right\|_{\infty}\leqslant
2\cdot\sqrt{3C_{d}\log N}\\}.$

This goes one step beyond mass concentration, as it asserts that on the
exponential scale, the dominant contribution to the integral
$M_{N}(b,\varepsilon)$ comes from functions for which the mass is relatively
evenly spread over the entire torus, we cannot have too many sharp peaks. This
phenomenon is closely related to (and proved by) an $\ell^{\infty}$ bound on
the associated free field.

######  Lemma 4.12.

Let $0<b\leqslant C_{d}$. We have for $n$ sufficiently large

$\operatorname{\mathds{P}}\left(\left\|\Psi^{\boldsymbol{0},L(b)}\right\|_{\infty}\geqslant\sqrt{3C_{d}\log
N}\right)\leqslant N^{-1}.$

###### Proof.

The translation invariance tells us that the random variables
$\left\\{\Psi^{\boldsymbol{0},L(b)}_{{\boldsymbol{x}}}\right\\}_{{\boldsymbol{x}}\in\mathds{T}^{d}}$
are identically distributed. The union bound is applicable and yields

$\displaystyle\operatorname{\mathds{P}}\left(\left\|\Psi^{\boldsymbol{0},L(b)}\right\|_{\infty}\geqslant\sqrt{3C_{d}\log
N}\right)$
$\displaystyle\leqslant\sum_{{\boldsymbol{x}}\in\mathds{T}^{d}}\operatorname{\mathds{P}}\left(\left|\Psi_{{\boldsymbol{x}}}^{\boldsymbol{0},L(b)}\right|^{2}\geqslant
3C_{d}\log N\right)$
$\displaystyle=\exp\left(\left(1-\frac{3C_{d}}{K^{\prime}_{n}(L(b))}\right)\cdot\log
N\right).$

The proof follows using the fact that $K^{\prime}_{N}(L(b))\to
K^{\prime}(L(b))\leqslant C_{d}$, and thus for $N$ sufficiently large,
$3C_{d}/K_{N}(L(b))\geqslant 2$. $\blacksquare$

###### Proof of Theorem 4.11.

Recall the definitions of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ be as in the
proof of Theorem 4.1. We define

$\mathcal{A}_{1}^{\prime}:=\mathcal{A}_{1}\cap\left\\{\psi:\left\|\psi^{\perp}\right\|_{\infty}\leqslant\sqrt{3C_{d}\log
N}\right\\}.$

It is clear that for $N$ sufficiently large,
$\mathcal{A}_{1}^{\prime}\subset\mathcal{A}^{\prime}$. We will denote

$\mathcal{A}_{\perp}:=\left\\{\psi^{\perp}\in\mathds{C}^{V}/\\{\phi^{\boldsymbol{0}}\\}:N(b^{\prime}-\varepsilon)\left\|\psi^{\perp}\right\|_{2}^{2}\leqslant
N(b^{\prime}+\varepsilon)\right\\}$

and

$\mathcal{A}_{\perp}^{\prime}:=\mathcal{A}_{\perp}\cap\left\\{\left\|\psi^{\perp}\right\|_{\infty}\leqslant\sqrt{3C_{d}\log
N}\right\\}.$

We use the fact that $\mathcal{A}_{2}$ contains $\mathcal{A}$ to conclude that
the ratio in (4.11) may be bounded below by

$\displaystyle\left(\int_{\mathcal{A}_{1}^{\prime}}\exp\left(-\left\|\nabla\psi\right\|\right)\mathbf{d}\psi\right)/\left(\int_{\mathcal{A}_{2}}\exp\left(-\left\|\nabla\psi\right\|\mathbf{d}\psi\right)\right).$
(35)

On separation, we then know that

$\int_{\mathcal{A}^{\prime}_{1}}\exp\left(\left\|-\nabla\psi\right\|_{2}^{2}\right)\mathbf{d}\psi=N\pi(b-b^{\prime}+\epsilon_{n})\cdot\int_{\mathcal{A}_{\perp}^{\prime}}\exp\left(-\left\|\nabla\psi^{\perp}\right\|_{2}^{2}\right)\mathbf{d}\psi^{\perp}.$

On introducing the exponential tilt, we further have

$\int_{\mathcal{A}_{1}^{\prime}}\exp\left(-\left\|\nabla\psi\right\|_{2}^{2}\right)\mathbf{d}\psi\geqslant
N\pi(b-b^{\prime}+\varepsilon)\cdot
e^{-L(b)(b-\varepsilon)}\cdot\int_{\mathcal{A}_{\perp}^{\prime}}\exp\left(-Q^{L(b)}(\psi^{\perp})\right)\mathbf{d}\psi^{\perp}.$

Analogously,

$\int_{\mathcal{A}_{2}}\exp\left(-\left\|\nabla\psi\right\|_{2}^{2}\right)\mathbf{d}\psi=N\pi(b+\varepsilon)\cdot
e^{L(b)(b+\varepsilon)}\cdot Z^{\boldsymbol{0},L(b)}.$

Thus (35) is bounded below further as

$\frac{b-b^{\prime}+\varepsilon}{b+\varepsilon}\cdot e^{-2\varepsilon
L(b)}\cdot\frac{1}{Z^{\boldsymbol{0},L(b)}}\int_{\mathcal{A}_{\perp}^{\prime}}\exp\left(-\left\|\nabla\psi\right\|_{2}^{2}\right)\mathbf{d}\psi.$

Now, observe that

$\displaystyle\frac{1}{Z^{\boldsymbol{0},y}}$
$\displaystyle\int_{\mathcal{A}_{\perp}^{\prime}}\exp\left(-\left\|\nabla\psi^{\perp}\right\|\right)\mathbf{d}\psi^{\perp}$
$\displaystyle=\operatorname{\mathds{P}}\left(b-\varepsilon\leqslant\frac{1}{N}\left\|\Psi^{\boldsymbol{0},L(b)}\right\|_{2}^{2}\leqslant
b+\varepsilon,\text{
}\left\|\Psi^{\boldsymbol{0},L(b)}\right\|_{\infty}\leqslant\sqrt{3C_{d}\log
N}\right)$ $\displaystyle\geqslant
1-\frac{cm_{N}(2)}{N\varepsilon^{2}}-\frac{1}{N}$

which follows via a combination of Lemmas 4.8 and 4.12, as well as the union
bound. $\blacksquare$

##  5\. Soliton Solutions and Minimal Energy

In this section, we provide a survey on some results describing soliton
solutions of the DNLS defined on the lattice $\mathds{Z}^{d}$. Recall, this
means that $\psi_{{\boldsymbol{x}}}(t)\in\ell^{2}(\mathds{Z}^{d})$ solves

$\displaystyle{\mathrm{i}\mkern
2.0mu}\frac{d}{dt}\psi_{{\boldsymbol{x}}}=-(\Delta\psi)_{{\boldsymbol{x}}}-|\psi_{{\boldsymbol{x}}}|^{p-1}\psi_{{\boldsymbol{x}}}.$

In particular, we will be interested in the time-periodic solutions, which we
refer to as discrete breathers or solitons.

###  5.1. Definition and Existence

There are two ways of characterizing soliton solutions. The first is finding
solutions via the ansatz $\psi_{{\boldsymbol{x}}}(t)=e^{{\mathrm{i}\mkern
2.0mu}\omega t}\varphi_{{\boldsymbol{x}}}$ where $\varphi_{{\boldsymbol{x}}}$
is time-invariant. The second is variational, solitons can be realized as the
minimizers of the Hamiltonian subject to the constraint
$\left\|\varphi\right\|_{2}^{2}=a$. The parameter $\omega$ is realized as a
Lagrange Multiplier and thus depends on $a$. The soliton equation is given by

$\displaystyle\omega(a)\cdot\varphi_{{\boldsymbol{x}}}=-(\Delta\phi)_{{\boldsymbol{x}}}-|\varphi_{{\boldsymbol{x}}}|^{p-1}\varphi_{{\boldsymbol{x}}}.$
(36)

The variational characterization also tells us that the values at all sites
must have the same complex phase, and thus the discrete solitons may be
assumed to be real-valued and nonnegative. Unlike the continuum case, there is
no scale invariance for the DNLS defined on a given lattice. Thus, we cannot
construct soliton solutions of arbitrary mass. The fundamental requirement for
(36) to admit an $\ell^{2}$ solution is that $\omega(a)<0$. As discussed in
the introduction, this occurs for all choices of mass $a>0$ when $p<1+4/d$, or
when $a>R_{p,d}$ for $p\geqslant 1+4/d$.

######  Lemma 5.1 (Weinstein, See [WEI99]).

The following holds.

  1. i.

Ground state or the minimizer in (11) exists when $I(a)\in(-\infty,0)$.

  2. ii.

Let $1<p<1+4/d$, then $I(a)<0$ for all $a>0$. Thus $R_{p}=0$.

  3. iii.

Let $p\geqslant 1+4/d$, then there exists a ground state excitation threshold
$R_{p}>0$ so that $I(a)<0$ if $a>R_{p}$; and $I(a)=0$ if $a<R_{p}$. Moreover,

$\displaystyle\frac{2}{p+1}R_{p}^{(p-1)/2}=\inf_{f}\left\\{\frac{\left\|f\right\|_{2}^{p-1}\cdot\left\|\nabla
f\right\|_{2}^{2}}{\left\|f\right\|_{p+1}^{p+1}}\right\\}.$

The last statement interprets $R_{p}$ in terms of a functional inequality for
the lattice. It is reciprocal to the best possible constant for the discrete
Gagliardo-Nirenberg-Sobolev inequality to hold [WEI99].

###  5.2. Dirichlet Solitons and Exponential Decay

In the continuum, when soliton solutions exist, they are known to be smooth
and exponentially decaying. In the discrete setting, exponential decay still
holds whenever the solitons exist, that is whenever $r>R_{p}$. A discrete
counterpart of the continuum proof can be used to prove this result on
$\mathds{Z}^{d}$. For our purposes, it suffices to establish a uniform rate of
exponential decay for the Dirichlet problem defined on a growing sequence of
boxes $\Lambda$ centered at the origin, and show the existence of an
exponentially decaying $\ell^{2}(\mathds{Z}^{d})$ minimizer via a tightness
argument. The analogous Dirichlet problem may be defined as

$\displaystyle\varphi^{\Lambda,a}:=\operatorname{argmin}\\{\mathcal{H}(\varphi):\varphi\in\mathds{C}^{\Lambda},\left\|\varphi\right\|_{2}^{2}=a\\}.$
(37)

It is clear that the Dirichlet solitons must satisfy

$\displaystyle\omega_{\Lambda}(a)\cdot\varphi^{\Lambda,a}_{{\boldsymbol{x}}}=\left(-\Delta^{0}_{\Lambda}\varphi^{\Lambda,a}\right)_{{\boldsymbol{x}}}-\left|\varphi^{\Lambda,a}_{{\boldsymbol{x}}}\right|^{p-1}\varphi^{\Lambda,a}_{{\boldsymbol{x}}}$
(38)

for a Lagrange multiplier $\omega_{\Lambda}(a)$. Note, minimizers for each
$\Lambda$ may be simultaneously defined on $\mathds{Z}^{d}$. Moreover, the
resulting sequence itself is minimizing.

######  Lemma 5.2.

Let $\varphi^{\Lambda,a}_{{\boldsymbol{x}}}$ denote the Dirichlet minimizers
of (37) with mass $a$ on $\Lambda\subset\mathds{Z}^{d}$. Let
$\\{\Lambda_{k}\\}_{k\geqslant 1}$ be a sequence of subsets each containing
$\boldsymbol{0}$ such that $\Lambda_{k}\uparrow\mathds{Z}^{d}$ as
$k\to\infty$. Then the sequence $\\{\varphi^{\Lambda_{k},a}\\}_{k\geqslant 1}$
is minimizing for (5).

###### Proof.

Let ${\varphi^{a,j}}$ be a minimizing sequence for $\mathcal{H}$, each with
mass $a$. Note, by the translation invariance of $\mathcal{H}$, we may
recenter the $\varphi^{a,j}$ such that $\boldsymbol{0}$ is always the site
with the largest absolute value. We then define

$\varphi^{a,j,\Lambda_{k}}:=\sqrt{\frac{a}{\left\|P_{\Lambda_{k}}\varphi^{a,j}\right\|_{2}^{2}}}\cdot
P_{\Lambda_{k}}\varphi^{a,j}.$

Note that as $\Lambda_{k}\uparrow\mathds{Z}^{d}$,
$P_{\Lambda_{k}}\varphi\to\varphi$ in norm for any
$\varphi\in\ell^{2}(\mathds{Z}^{d})$. We may then choose a diagonal
subsequence that is minimizing for $H$ and denote it as
$\\{\varphi^{a,j_{k},\Lambda_{l}}\\}_{k\geqslant 1}$. By hypothesis,

$\mathcal{H}(\varphi^{\Lambda_{k},a})\leqslant\mathcal{H}(\varphi^{n_{k},a,\Lambda_{k}}).$

Thus, the sequence of Dirichlet minimizers is also minimizing for the
Hamiltonian. $\blacksquare$

On the question of exponential decay, we provide a probabilistic proof
directly adapted from Chatterjee [C14], who proved the same for soliton
solutions in the mass subcritical regime on the discrete torus.

We bring this in now in order to control the value of the Lagrange Multiplier,
which in turn is required for a uniform rate of exponential decay.

######  Lemma 5.3.

Let $a>R_{p}$, for $|\Lambda|$ sufficiently large we have
$\varepsilon_{1}<\varepsilon_{2}<0$ depending on $a$ such that

$\varepsilon_{1}<\omega_{\Lambda}(a)<\varepsilon_{2}.$

###### Proof.

Multiplying both sides of (38) by $\varphi^{\Lambda,a}_{{\boldsymbol{x}}}$
(recall that $\varphi^{\Lambda,a}$ is assumed to be nonnegative), we obtain
that

$\displaystyle\omega_{\Lambda}(a)\cdot r$
$\displaystyle=\left\|\nabla\varphi^{\Lambda,a}\right\|_{2}^{2}-\left\|\varphi^{\Lambda,a}\right\|_{p+1}^{p+1}$
$\displaystyle\leqslant\left\|\nabla\varphi^{\Lambda,a}\right\|_{2}^{2}-\frac{2}{p+1}\left\|\varphi^{\Lambda,a}\right\|_{p+1}^{p+1}=\mathcal{H}(\varphi^{\Lambda,a}).$

Now recall that as $\Lambda\uparrow\mathds{Z}^{d}$,
$\mathcal{H}(\varphi^{\Lambda,a})$ converges to $I(a)<0$, and we choose
$\varepsilon_{2}>I(a)$ but still strictly negative. As for the lower bound, we
have that $\omega_{\Lambda}(a)\geqslant-a^{(p+1)/2}=:\varepsilon_{1}.$
$\blacksquare$

######  Lemma 5.4.

Let $a>R_{p}$. Let $\varphi^{\Lambda}$ denote a Dirichlet minimizer
corresponding to $\Lambda$. We have $U_{0}\subset\Lambda$ and finite constants
$\omega_{0}>0$ and $C_{0}>0$ depending on $a$ and independent of $\Lambda$
such that

$\displaystyle\varphi^{\Lambda}_{{\boldsymbol{x}}}\leqslant
C_{0}e^{-\omega_{0}\cdot d({\boldsymbol{x}},U_{0})}.$

###### Proof.

As we have already seen, $a>R_{p}$ implies that for $I(a)<\varepsilon<0$,
$\omega_{\Lambda}(a)<\varepsilon$ for all $|\Lambda|$ sufficiently large. In
turn, (38) may be rewritten using Green’s function description of the inverse
of the Dirichlet Laplacian as

$\displaystyle\varphi^{\Lambda,a}_{{\boldsymbol{x}}}=\sum_{{\boldsymbol{x}^{\prime}}}G^{\Lambda}_{-\omega_{\Lambda}}({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})\left|\varphi^{\Lambda,a}_{{\boldsymbol{x}^{\prime}}}\right|^{p-1}\varphi^{\Lambda,a}_{{\boldsymbol{x}^{\prime}}}.$
(39)

Now, let $\delta>0$, and define

$U_{\delta}:=\\{{\boldsymbol{x}}\in\Lambda:\varphi^{\Lambda,a}_{{\boldsymbol{x}}}\geqslant\delta\\}.$

By the usual bound,

$|U_{\delta}|\leqslant\frac{a}{\delta^{2}}.$

We define $z=\omega_{\Lambda}(a)^{2}/(1+\omega_{\Lambda}(a)^{2})$, and let
$(X^{z}_{t})_{t\in\mathds{N}}$ be the simple symmetric random walk on
$\Lambda$, started at ${\boldsymbol{x}}$ which is killed with probability $z$
at each step, and is killed at the boundary. Recall that the Green’s function
is given by

$G^{\Lambda}_{|\omega_{\Lambda}|}({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})=\operatorname{\mathds{E}}\sum_{t=1}^{T_{\Lambda}^{|\omega_{\Lambda}|}}\mathds{1}_{X_{t}^{z}={\boldsymbol{x}^{\prime}}}.$

Using this expression and the fact that the event of death is independent of
the step taken, we may rewrite (39) as

$\varphi^{\Lambda,a}_{{\boldsymbol{x}}}=\sum_{t=0}^{\infty}(1-z)^{t}\operatorname{\mathds{E}}\left|\varphi^{\Lambda,a}_{X_{t}^{z}}\right|^{p-1}\varphi^{\Lambda,a}_{X_{t}^{z}}.$

Let $p_{\Lambda}({\boldsymbol{x}},{\boldsymbol{x}^{\prime}},t)$ denote the
probability kernel of $X_{t}$, the random walk on $\Lambda$ annihilated on the
boundary. Observe that for all ${\boldsymbol{x}^{\prime}}$ such that
$d({\boldsymbol{x}^{\prime}},{\boldsymbol{x}})>t$, the probability that the
random walk has reached ${\boldsymbol{x}^{\prime}}$ is $0$. Thus,

$\displaystyle\varphi^{\Lambda,a}_{{\boldsymbol{x}}}=\sum_{{\boldsymbol{x}^{\prime}}\in\Lambda}\sum_{t\geqslant
d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})}(1-z)^{t}p_{\Lambda}({\boldsymbol{x}},{\boldsymbol{x}^{\prime}},t)\left(\varphi^{\lambda,a}_{{\boldsymbol{x}^{\prime}}}\right)^{p}.$
(40)

Clearly $p({\boldsymbol{x}},\mathbf{y},t)\leqslant 1$ and on evaluation of the
geometric sum, we have

$\varphi^{\Lambda,a}_{{\boldsymbol{x}}}\leqslant\frac{1}{z}\sum_{{\boldsymbol{x}^{\prime}}\in\Lambda}{(1-z)^{d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})}}\left(\varphi^{\lambda,a}_{{\boldsymbol{x}^{\prime}}}\right)^{p}.$

Now, if ${\boldsymbol{x}^{\prime}}\notin U_{\delta}$, then we have that
$(\varphi^{\Lambda,a}_{{\boldsymbol{x}}})^{p}\leqslant\delta^{p-1}\varphi_{{\boldsymbol{x}^{\prime}}}^{\Lambda,a}$,
and if ${\boldsymbol{x}^{\prime}}\in U_{\delta}$, then
$d({\boldsymbol{x}},U_{\delta})\leqslant
d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})$. Thus, partitioning the sum in
(40),

$\displaystyle\varphi^{\lambda,a}_{{\boldsymbol{x}}}$
$\displaystyle\leqslant\frac{(1-z)^{d(U_{\delta},{\boldsymbol{x}})}}{z}\sum_{{\boldsymbol{x}^{\prime}}\in
U_{\delta}}\left(\varphi^{\Lambda,a}_{{\boldsymbol{x}^{\prime}}}\right)^{p}+\frac{\delta^{p-1}}{z}\sum_{{\boldsymbol{x}^{\prime}}\notin
U_{\delta}}\varphi^{\Lambda,a}_{{\boldsymbol{x}^{\prime}}}$
$\displaystyle\leqslant\frac{(1-z)^{d(U_{\delta},{\boldsymbol{x}})}}{z}\cdot
a^{p/2}\cdot\frac{a}{\delta^{2}}+\frac{\delta^{p-1}}{z}\sum_{{\boldsymbol{x}^{\prime}}\in\Lambda}(1-z)^{d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})}\varphi^{\Lambda,a}_{{\boldsymbol{x}^{\prime}}}.$
(41)

Let

$\eta_{{\boldsymbol{x}}}:=\frac{a^{(p+2)/2}}{z\delta^{2}}\cdot\max_{{\boldsymbol{x}^{\prime}}^{\prime}\in\Lambda}(1-z)^{d({\boldsymbol{x}^{\prime}}^{\prime},U_{\delta})+d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}}^{\prime})/2}.$

We observe as a consequence of the triangle inequality,

$\eta_{{\boldsymbol{x}}}\leqslant\frac{a^{(p+2)/2}}{z\delta^{2}}\max_{{\boldsymbol{x}^{\prime}}^{\prime}\in\Lambda}(1-z)^{d({\boldsymbol{x}^{\prime}}^{\prime},U_{\delta})+d({\boldsymbol{x}^{\prime}},{\boldsymbol{x}^{\prime}}^{\prime})/2-d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})/2}\leqslant(1-z)^{-d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})}\eta_{{\boldsymbol{x}^{\prime}}}.$

Let $C$ denote the smallest constant such that

$\displaystyle\varphi^{\Lambda,a}_{{\boldsymbol{x}}}\leqslant
C\eta_{{\boldsymbol{x}}}.$ (42)

Since $\eta_{{\boldsymbol{x}}}>0$ and $\Lambda$ is finite, it is clear that
$C$ is finite as well. We use this bound for (41) obtaining

$\displaystyle\varphi^{\Lambda,a}_{{\boldsymbol{x}}}$
$\displaystyle\leqslant\frac{(1-z)^{d(U_{\delta},{\boldsymbol{x}})}}{z}\cdot\frac{a^{(p+2)/2}}{\delta^{2}}+\frac{C\delta^{p-1}}{z}\sum_{{\boldsymbol{x}^{\prime}}\in\Lambda}(1-z)^{d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})}\eta_{{\boldsymbol{x}^{\prime}}}$
$\displaystyle\leqslant\frac{(1-z)^{d(U_{\Delta},{\boldsymbol{x}})}}{z}\cdot\frac{a^{(p+2)/2}}{\delta^{2}}+\frac{C\delta^{p-1}}{z}\eta_{{\boldsymbol{x}}}\sum_{{\boldsymbol{x}^{\prime}}\in\Lambda}(1-z)^{\frac{1}{2}d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})}$
$\displaystyle\leqslant\left(1+C\cdot\delta^{p-1}\sum_{{\boldsymbol{x}^{\prime}}\in\Lambda}(1-z)^{d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})/2}\right)\varphi_{{\boldsymbol{x}}}.$

We can choose $\delta$ sufficiently small, such that

$\frac{\delta^{p-1}}{z}\sum_{{\boldsymbol{x}^{\prime}}\in\Lambda}(1-z)^{d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})}\leqslant\frac{1}{2}.$

In turn, this tells us that

$C\leqslant 1+{C}/{2}$

since $C$ is the best constant for (42). This is the same as $C\leqslant 2$.
With this choice of $\delta$, we have that

$\varphi^{\Lambda,a}_{{\boldsymbol{x}}}\leqslant
2\frac{a^{(p+2)/2}}{z\delta^{2}}\max_{{\boldsymbol{x}^{\prime}}\in\Lambda}(1-z)^{d(U_{\delta},{\boldsymbol{x}^{\prime}})+d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})/2}.$

By the triangle inequality,

$d(U_{\delta},{\boldsymbol{x}^{\prime}})+\frac{1}{2}d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})\geqslant\frac{1}{2}d(U_{\delta},{\boldsymbol{x}^{\prime}})+\frac{1}{2}d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}})\geqslant\frac{d(U_{\delta},{\boldsymbol{x}})-d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}}^{\prime})}{2}+\frac{1}{2}d({\boldsymbol{x}},{\boldsymbol{x}^{\prime}}).$

Thus,

$\varphi^{\Lambda,a}_{{\boldsymbol{x}}}\leqslant
2\frac{a^{(p+2)/2}}{z\delta^{2}}(1-z)^{d({\boldsymbol{x}},U_{\delta})/2}.$

To conclude, take $U_{0}=U_{\delta},\text{
}C_{0}=2\frac{a^{(p+2)/2}}{z\delta^{2}}\text{ and
}\omega_{0}=-\frac{1}{2}\log(1-z).$ $\blacksquare$

We take this moment to emphasize that we may bound all the constants arising
in Lemma 5.4 in terms of the $\varepsilon_{1}$ and $\varepsilon_{2}$
introduced in Lemma 5.3. To begin with, as a direct consequence of Lemma 5.3,

$\frac{\varepsilon^{2}_{2}}{1+\varepsilon^{2}_{2}}\leqslant
z\leqslant\frac{\varepsilon^{2}_{1}}{1+\varepsilon^{2}_{1}}.$

Next, a valid choice of $\delta$ is

$\delta=\frac{z|\log(1-z)|^{d}}{2(d-1)!}$

which in turn yields

$C_{0}=\frac{4\cdot
a^{(p+1)/2}\cdot(d-1)!}{z^{2}|\log(1-z)|^{d}}\leqslant\frac{4\cdot
a^{(p+1)/2}\cdot(d-1)!\cdot(1+\varepsilon_{2}^{2})}{\varepsilon_{2}^{2}\cdot|\log(1+\varepsilon^{2}_{2})-2\log\varepsilon_{2}|^{d}}$

The uniform rate of exponential decay of the Dirichlet minimizers implies the
existence of an exponentially decaying minimizer to (5), via a standard
compactness argument. For the sake of clarity of exposition, we detail the
argument here.

######  Lemma 5.5.

Let $a>R_{p}$. There exists an exponentially decaying minimizer $\varphi^{a}$
to (5).

###### Proof.

Given a box $\Lambda=[-n,n]^{d}\cap\mathds{Z}^{d}$ centered at the origin, we
define $2\cdot\Lambda$ as $[-2n,2n]^{d}\cap\mathds{Z}^{d}$. We embed the
minimizer $\varphi^{\Lambda,a}$ into $\ell^{2}(2\cdot\Lambda)$ via the
standard inclusion map and note that the energy is still the same. We are then
free to translate $\varphi^{\Lambda,a}$ within this larger box and preserve
the energy. We define $\tilde{\varphi}^{\Lambda,a}$ to be the translation of
$\varphi^{\Lambda,a}$ such that the site with the largest mass is situated at
the origin. By Lemma 5.4 there exist, independent of $\Lambda$, a set $U_{0}$
of bounded size and positive constants $C_{0}$ and $\omega_{0}$ such that
$\tilde{\varphi}^{\Lambda,a}\leqslant
C_{0}e^{-\omega_{0}d({\boldsymbol{x}},U_{0})}$. By construction, it is clear
that $\boldsymbol{0}\in U_{0}$, and thus there is a box $\Lambda_{0}$ such
that $\tilde{\varphi}^{\Lambda,a}\leqslant C_{0}e^{-\omega_{0}\cdot
d(\Lambda_{0},{\boldsymbol{x}})}$. Thus, $\\{\tilde{\varphi}^{\Lambda,a}\\}$
is a pre-compact sequence in $\ell^{2}(\mathds{Z}^{d})$. We take $\varphi^{a}$
to be any accumulation point of the sequence. Clearly,
$\varphi^{a}_{{\boldsymbol{x}}}\geqslant 0$,
$\left\|\varphi^{a}\right\|_{2}^{2}=a$ and
$\varphi^{a}_{{\boldsymbol{x}}}\leqslant
C_{0}e^{-\omega_{0}d({\boldsymbol{x}},\Lambda_{0})}$. $\blacksquare$

######  Remark 5.6.

We remark that the above process of re-centering should not be necessary as
the sequence of Dirichlet minimizers should have a mass that is concentrated
towards the center of the box $\Lambda$. This is due to the decay of the
Dirichlet heat kernel at the edges of the box. We, in fact, conjecture that
the minimizer for the Dirichlet problem is unique, which would in turn imply
the uniqueness of the minimizer to (5) up to translation and phase rotation.

The exponentially decaying minimizer and the corresponding truncations are
very useful from the perspective of constructing a minimizing sequence with a
good rate of convergence. We establish this now.

######  Lemma 5.7.

Let $a>R_{p}+\varepsilon$, $\Lambda=[-n,n]^{d}\cap\mathds{Z}^{d}$, and let
$\varphi^{\Lambda,a}$ denote the Dirichlet minimizer. We have

$\left|I(a)-\mathcal{H}\left(\varphi^{\Lambda,a}\right)\right|\leqslant
C_{0}^{\prime}\cdot a^{p^{2}-1}\cdot e^{-\omega_{0}\cdot n}.$

###### Proof.

Let $\varphi^{a}$ be as in Lemma 5.5. Note that for any $p$,

$\displaystyle\left\|\varphi^{a}\right\|^{p}_{p}-\left(\frac{a}{\left\|P_{\Lambda}\varphi^{r}\right\|_{2}^{2}}\right)^{p/2}\left\|P_{\Lambda}\varphi^{a}\right\|_{p}^{p}\leqslant\left|1-\left(\frac{a}{\left\|P_{\Lambda}\varphi^{r}\right\|_{2}^{2}}\right)^{p}\right|\left\|\varphi^{a}\right\|_{p}^{p}.$
(43)

By construction, we know that for all ${\boldsymbol{x}}$ outside $\Lambda$

$\varphi^{a}_{{\boldsymbol{x}}}\leqslant C_{0}\exp(-\omega_{0}\cdot n).$

Thus, we have that

$\left\|P_{\Lambda}\varphi^{a}\right\|_{2}^{2}\geqslant
a-\frac{C_{0}\cdot\Gamma(d)}{\omega_{0}^{d}}\exp\left({-\omega_{0}\cdot
n}\right).$

For the convenience of notation, we will introduce for the purposes of this
proof

$\epsilon(n):=C_{0}\exp(-\omega_{0}\cdot n).$

Applied to (43),

$\displaystyle\left\|\varphi^{a}\right\|^{p+1}_{p+1}-\left(\frac{a}{\left\|P_{\Lambda}\varphi^{a}\right\|_{2}^{2}}\right)^{(p+1)/2}\left\|P_{\Lambda}\varphi^{a}\right\|_{p+1}^{p+1}$
$\displaystyle\leqslant\left|\frac{a^{p+1}-(a-\epsilon(n))^{p+1}}{(a-\epsilon(n))^{p}}\right|\left\|\varphi^{a}\right\|^{p}_{p}$
$\displaystyle\leqslant\frac{(p+1)a^{p(p+1)}\epsilon(n)}{(a-\epsilon(n))^{p+1}}.$

Now as for the gradient term, we only have to control the contributions from
outside $\Lambda$ and on the boundary. We have that

$\displaystyle\left\|\nabla\varphi^{a}\right\|^{2}_{2}-\frac{a}{\left\|P_{\Lambda}\varphi^{a}\right\|_{2}^{2}}\cdot\left\|\nabla
P_{\Lambda}\varphi^{a}\right\|_{2}^{2}\leqslant\sum_{{\boldsymbol{x}}\sim{\boldsymbol{x}^{\prime}}\in\Lambda^{c}}|\varphi^{a}_{{\boldsymbol{x}}}-\varphi^{a}_{{\boldsymbol{x}^{\prime}}}|^{2}$
$\displaystyle+\sum_{{\boldsymbol{x}}\sim{\boldsymbol{x}^{\prime}}\in\Lambda}\left(1-\frac{a}{\left\|P_{\Lambda}\varphi^{a}\right\|_{2}^{2}}\right)\cdot|\varphi^{a}_{{\boldsymbol{x}}}-\varphi^{a}_{{\boldsymbol{x}^{\prime}}}|^{2}$
$\displaystyle+\sum_{{\boldsymbol{x}}\sim{\boldsymbol{x}^{\prime}}\in\partial\Lambda}|\varphi^{a}_{{\boldsymbol{x}}}|^{2}.$

Applying the bound obtained from the exponential decay,

$\left\|\nabla\varphi^{a}\right\|^{2}_{2}-\frac{a}{\left\|P_{\Lambda}\varphi^{a}\right\|_{2}^{2}}\cdot\left\|\nabla
P_{\Lambda}\varphi^{a}\right\|_{2}^{2}\leqslant
2\epsilon(n)+\left(\frac{\epsilon(n)}{a-\epsilon(n)}\right)a.$

Combining, we have

$\left|I(a)-\mathcal{H}\left(\sqrt{\frac{a}{\left\|P_{\Lambda}\varphi^{a}\right\|}}\varphi^{a}\right)\right|\leqslant\epsilon(n)\cdot\left(\frac{a^{p(p+1)}}{2(a-\epsilon(n))^{p+1}}+\frac{2a}{a-\epsilon(n)}\right)\leqslant
C_{0}^{\prime}\cdot a^{p^{2}-1}\cdot e^{-\omega_{0}\cdot n}.$

To conclude, we note that

$I(a)\leqslant\mathcal{H}(\varphi^{\Lambda,a})\leqslant\mathcal{H}\left(\sqrt{\frac{a}{\left\|P_{\Lambda}\varphi^{a}\right\|_{2}^{2}}}P_{\Lambda}\varphi^{a}\right).$

This completes the proof. $\blacksquare$

For the scenario where the minimizer does not exist, any weakly convergent
sequence to zero is minimizing. We may use this fact to bound the rate of
convergence.

######  Lemma 5.8.

Let $a\leqslant R_{p}$, $\Lambda=[-n,n]^{d}\cap\mathds{Z}^{d}$ and
$\varphi^{\Lambda,a}$ be the Dirichlet minimizer with mass $a$. Then we have

$|\mathcal{H}(\varphi^{\Lambda,a})|\leqslant\frac{2ad}{|\Lambda|^{1/d}}.$

###### Proof.

Let $\boldsymbol{1}_{\Lambda,{\boldsymbol{x}}}$ denote the function which
takes the value $1$ for every ${\boldsymbol{x}}\in\Lambda$ and is zero outside
$\Lambda$. It is easy to evaluate the norms of $\boldsymbol{1}_{\Lambda}$, in
particular

$\left\|\nabla\boldsymbol{1}_{\Lambda}\right\|_{2}^{2}\leqslant
2d|\Lambda|^{(d-1)/d}.$

Thus,

$\mathcal{H}\left(\sqrt{\frac{a}{|\Lambda|}}\boldsymbol{1}_{\Lambda}\right)\leqslant\frac{2ad}{|\Lambda|^{1/d}}.$

Since we are working with $a\leqslant R_{p}$, we know that $I(a)=0$. Combined
with the fact that $\mathcal{H}(\varphi^{\Lambda,a})$ is monotonically
decreasing, we must have that $\mathcal{H}(\varphi^{\Lambda,a})\geqslant 0$.
For $|\Lambda|$ sufficiently large, by the minimizing hypothesis, we have

$0\leqslant\mathcal{H}(\varphi^{\Lambda,a})\leqslant\mathcal{H}\left(\sqrt{\frac{a}{|\Lambda|}}\boldsymbol{1}_{\Lambda}\right)\leqslant\frac{2ad}{|\Lambda|^{1/d}}$

and this completes the proof. $\blacksquare$

###  5.3. Analysis of Minimal Energy

We conclude this section with some basic facts about the function $I$.
Clearly, we have

$\left\|\nabla\psi\right\|_{2}^{2}\leqslant
4d\left\|\psi\right\|_{2}^{2}\text{ and
}\left\|\psi\right\|_{p+1}^{p+1}\leqslant\left\|\psi\right\|_{2}^{2}\cdot\max_{x\in\mathds{Z}^{d}}|\psi_{x}|^{p-1}\leqslant\left\|\psi\right\|_{2}^{p+1}.$

Define the functions $J,\widehat{J}:(0,\infty)\to[0,\infty)$ by

$\displaystyle J(a):=\frac{1}{a}I(a)\text{ and
}\widehat{J}(a):=\frac{1}{a}I(a)+\frac{2}{p+1}(a\vee R_{p})^{(p-1)/2}.$ (44)

######  Lemma 5.9.

The function $J$ is decreasing and differentiable.

###### Proof.

Fix $0<a^{\prime}<a$. Given a function $\psi$ with mass $a^{\prime}$, we
consider the function $\tilde{\psi}(x)=\sqrt{a/a^{\prime}}\cdot\psi(x)$ with
mass $a$. We have

$\displaystyle\mathcal{H}(\tilde{\psi})=\frac{a}{a^{\prime}}\left\|\nabla\psi\right\|_{2}^{2}-\frac{2}{p+1}\left(\frac{a}{a^{\prime}}\right)^{(p+1)/2}\left\|\psi\right\|_{p+1}^{p+1},$

or

$\frac{I(a)}{a}\leqslant\frac{1}{a}\mathcal{H}(\tilde{\psi})=\frac{1}{a^{\prime}}\mathcal{H}(\psi)-2\cdot\frac{a^{(p-1)/2}-(a^{\prime})^{(p-1)/2}}{(p+1)\cdot(a^{\prime})^{(p+1)/2}}\left\|\psi\right\|_{p+1}^{p+1}\leqslant\frac{\mathcal{H}(\psi)}{a^{\prime}}.$

Thus we have $\frac{I(a)}{a}\leqslant\frac{I(a^{\prime})}{a^{\prime}}$.
Similarly, given a function $\tilde{\psi}$ with mass $a$, we consider the
function $\psi(x)=\sqrt{a^{\prime}/a}\cdot\tilde{\psi}(x)$ with mass $s$ to
have

$\displaystyle\frac{I(a^{\prime})}{a^{\prime}}\leqslant\frac{1}{a^{\prime}}\mathcal{H}(\psi)$
$\displaystyle=\frac{1}{a}\mathcal{H}(\tilde{\psi})+\frac{a^{(p-1)/2}-(a^{\prime})^{(p-1)/2}}{(p+1)a^{(p+1)/2}}\left\|\tilde{\psi}\right\|_{p+1}^{p+1}$
$\displaystyle\leqslant\frac{1}{a}\mathcal{H}(\tilde{\psi})+\frac{2}{p+1}(a^{(p+1)/2}-(a^{\prime})^{(p-1)/2}).$

Thus we have

$0\leqslant\frac{I(a^{\prime})}{a^{\prime}}-\frac{I(a)}{a}\leqslant\frac{2}{p+1}(a^{(p-1)/2}-(a^{\prime})^{(p-1)/2})$

and the proof is complete. $\blacksquare$

A trivial corollary of this result is the differentiability of the $I$
function itself, merely a consequence of the product rule.

######  Lemma 5.10.

$\widehat{J}$ is increasing in $r$. Moreover,

$\frac{2}{p+1}R_{p}^{(p-1)/2}\leqslant\widehat{J}(a)\leqslant 1/C_{d}\text{
for all }a>0.$

###### Proof.

By evaluating $\mathcal{H}$ at the function
$\psi(x)=\sqrt{a}\cdot\mathds{1}_{x=0}$ in (11), it is clear that

$\displaystyle I(a)\leqslant 2d\cdot a-\frac{2}{p+1}a^{(p+1)/2}.$ (45)

From Lemma 5.9, we have for $a\geqslant R_{p}$ and $a^{\prime}=R_{p}$

$\displaystyle I(a)\geqslant\frac{2}{p+1}R_{p}^{(p-1)/2}\cdot
a-\frac{2}{p+1}a^{(p+1)/2}.$ (46)

$\blacksquare$

##  6\. Convergence of the Free Energy

In this section, we will establish the convergence of the scaled free energy.
Recall that our measure is restricted to the ball of radius $\sqrt{N}$ in
$\mathds{C}^{V}$. We will be cutting the mass constraint into pieces
corresponding to “solitonic” and dispersive contributions. The vertex set $V$
will be partitioned into a set $U$ which is small, where a typical function is
concentrated, and $U^{c}$ where the mass of a typical function is spread out
over all the sites. Recall that we denote the restrictions of $\psi$ onto $U$
and $U^{c}$ as $\psi_{U}$ and $\psi_{U^{c}}$ respectively. We denote the
collection of sites ${\boldsymbol{x}}\in U$ adjacent to $U^{c}$ by $\partial
U_{int}$ and analogously the collection of sites ${\boldsymbol{x}^{\prime}}\in
U^{c}$ adjacent to $U$ by $\partial U_{ext}$. The Hamiltonian
$\mathcal{H}_{n}$ may then be written as

$\displaystyle\begin{split}\mathcal{H}_{n}(\psi)&=\mathcal{H}_{n}(\psi_{U})+\mathcal{H}_{n}(\psi_{U^{c}})\\\
&\qquad+\sum_{{\boldsymbol{x}}\sim{\boldsymbol{x}^{\prime}}\in\partial
U}|\psi_{{\boldsymbol{x}}}-\psi_{{\boldsymbol{x}^{\prime}}}|^{2}-\sum_{{\boldsymbol{x}}\in\partial
U_{int}}|\psi_{{\boldsymbol{x}}}|^{2}-\sum_{{\boldsymbol{x}^{\prime}}\in\partial
U_{ext}}|\psi_{{\boldsymbol{x}^{\prime}}}|^{2}.\end{split}$ (47)

In the definitions of $\mathcal{H}_{n}(\psi_{U})$ and
$\mathcal{H}_{n}(\psi_{U^{c}})$, the Laplacian is understood to mean the
Dirichlet Laplacian. The decomposition of $\psi$ into $\psi_{U}$ and
$\psi_{U^{c}}$ is to be regarded as the orthogonal decomposition
$\mathds{C}^{V}=\mathds{C}^{U}\oplus\mathds{C}^{U^{c}}$, and as such the
expression $\psi=\psi_{U}\oplus\psi_{U^{c}}$ carries the obvious meaning, as
does the decomposition of the volume element
$\mathbf{d}\psi=\mathbf{d}\psi_{U}\cdot\mathbf{d}\psi_{U^{c}}$. The mass
constraint, which may be expressed as

$\left\|\psi_{U}\right\|_{2}^{2}+\left\|\psi_{U^{c}}\right\|_{2}^{2}\leqslant
N,$

cannot be expressed as a product set, but can be closely approximated by a
disjoint union of product sets. For a choice of $0<\alpha<1$ to be specified
later, let $\kappa=N^{\alpha}$. Let $i^{*}$ be the smallest natural number
such that $i^{*}\kappa\geqslant N$ and $i_{*}$ be the largest natural number
such that $i_{*}\kappa\leqslant N$. We have that

$\displaystyle\bigcup_{i+j\leqslant
i^{*}}\left\\{\psi_{U}:\left\|\psi_{U}\right\|_{2}^{2}\in[i\kappa,i\kappa+\kappa)\right\\}\times\left\\{\psi_{U^{c}}:\left\|\psi_{U^{c}}\right\|_{2}^{2}\in[j\kappa,(j+1)\kappa)\right\\}$
(48)

contains the ball of radius $\sqrt{N}$, and

$\displaystyle\bigcup_{i+j=i_{*}}\left\\{\psi_{U}:\left\|\psi_{U}\right\|_{2}^{2}\in[i\kappa,i\kappa+\kappa)\right\\}\times\left\\{\psi_{U^{c}}:\left\|\psi_{U^{c}}\right\|_{2}^{2}\in[j\kappa,(j+1)\kappa)\right\\}$
(49)

is contained by the ball. Replacing the ball with these sets as the region of
integration will give us upper and lower bounds for the partition function.
For ease of notation in the sections to follow, we define

$\displaystyle\mathcal{B}_{i,U}$
$\displaystyle=\left\\{\psi_{U}:\left\|\psi_{U}\right\|_{2}^{2}\in[i\kappa,i\kappa+i)\right\\}$
(50) $\displaystyle\text{and }\mathcal{A}_{i,j,U}$
$\displaystyle=\mathcal{B}_{i,U}\times\mathcal{B}_{j,U^{c}}.$

To summarize, for a given fixed $U\subset V$, we have

$\displaystyle\bigcup_{i+j=i_{*}}\mathcal{A}_{i,j,U}\subset\left\\{\psi:\left\|\psi\right\|_{2}^{2}<N\right\\}\subset\bigcup_{i+j\leqslant
i^{*}}\mathcal{A}_{i,j,U}.$ (51)

###  6.1. Upper Bound

With the decomposition of the mass constraint introduced above, we will split
the proof of Theorem 2.3 into two parts. This section is dedicated to the
first portion, the appropriate upper bound for the free energy.

######  Lemma 6.1.

We have a constant $C$ and a sequence $\Theta_{S}(N)$ depending on $\nu$,
$\theta$, $p$ and $d$ such that

$\displaystyle\frac{1}{N}\log
Z_{N}(\theta,\nu)\leqslant\log(\pi/\theta)-\inf_{0<a<1}(W(\theta(1-a))+\theta\nu^{-1}I(a\nu))+\frac{1}{N}\log\Theta_{S}(N)$

where

$\Theta_{S}(N)\leqslant\exp\left(10s_{N}N\right)\cdot N^{2s_{N}^{-d-1}}\cdot
e^{C\kappa_{N}}.$

We begin with an argument to justify the separation of functions into
concentrated and dispersed parts, in other words showing that for any function
$\psi\in\mathds{C}^{V}$, we may always find a set $U$ such that $\psi$ is not
concentrated outside $U$, and the boundary contribution may be controlled.

######  Lemma 6.2.

Given a function $\psi\in\mathds{C}^{V}$ with $\left\|\psi\right\|_{2}^{2}<N$
and $\varepsilon>0$, there exists a subset $U\subset V$ with
$|U|\leqslant\varepsilon^{-d-1}$ such that $|\psi_{x}|^{2}<\varepsilon N$ for
all $x\notin U$ and $\sum_{x\in\partial U_{int}\cup\partial
U_{ext}}|\psi_{x}|^{2}<3\varepsilon N$.

###### Proof.

Take $U_{0}=\\{\boldsymbol{x}:|\psi_{\boldsymbol{x}}|^{2}\geqslant\varepsilon
N\\}$. Clearly $|U_{0}|\leqslant 1/\varepsilon$. We define the $U_{i}$ by the
successive addition of the 2-step outer boundary, _i.e.,_
$U_{i}=\\{\boldsymbol{x}\in V:d(\boldsymbol{x},U_{0})=2i\\}$ and
$B_{i}=U_{i}\setminus U_{i-1}$ for all $i\geqslant 1$. Take $B_{0}=U_{0}$.
Note that

$\sum_{i=1}^{k}\sum_{\begin{subarray}{c}\boldsymbol{x}\in
B_{i}\end{subarray}}|\psi_{\boldsymbol{x}}|^{2}\leqslant\sum_{\boldsymbol{x}}|\psi_{\boldsymbol{x}}|^{2}\leqslant
N.$

In particular, there exists $i\leqslant k$ such that $\sum_{\boldsymbol{x}\in
B_{i}}|\psi_{\boldsymbol{x}}|^{2}\leqslant N/k$. Take
$k=\lfloor\varepsilon^{-1}/2\rfloor$, so that $N/k\leqslant 10\varepsilon N$.
Now, note that $|U_{i}|\leqslant(2k+1)^{d}|U_{0}|\leqslant\varepsilon^{-d-1}$.
$\blacksquare$

We will refer to the set $U$ as $\varepsilon-$admissible for $\psi$, and what
Lemma 6.2 states that given a function $\psi$ with appropriate mass and
$\varepsilon>0$, we may find an $\varepsilon-$admissible set. For a fixed set
$U$ and a positive sequence $s_{N}$ decaying to zero at a specified rate (see
(74)), we define

$\displaystyle\mathcal{U}(s_{N}):=\\{\psi\in\mathds{C}^{n}:U\text{ is
$s_{N}-$admissible for $\psi$}\\}.$

It is easy to see that if $\mathcal{U}$ is non-empty, then it must be open in
$\mathds{C}^{V}$ since we may perturb slightly around any $\psi$ that is
contained. Therefore, if it is non-empty, it must have a non-zero Lebesgue
measure. Further, for every $\psi$ with appropriate mass, we know that there
must be a $U$ with size bounded above by $s_{N}^{-d-1}$ such that $U$ is
$s_{N}$ admissible for $\psi$. Thus, we have that

$\\{\psi:\left\|\psi\right\|_{2}^{2}<N\\}\subset\bigcup_{|U|\leqslant
s_{N}^{-d-1}}\mathcal{U}(s_{N}).$

On combination with (51), we have

$\displaystyle Z_{N}\leqslant\sum_{|U|\leqslant
s_{N}^{-d-1}}\mathcal{U}(s_{N})\sum_{i+j\leqslant
i^{*}}\int_{\mathcal{U}\cap\mathcal{A}_{i,j,U}}\exp(-\mathcal{H}_{N}(\psi))\mathbf{d}\psi.$
(52)

With a fixed choice of $U$ and admissible $\psi$, the bound on the gradient
and (47) yield

$\displaystyle\exp(-\mathcal{H}_{N}(\psi))\leqslant\exp(-\mathcal{H}_{N}(\psi_{U}))\cdot\exp(-\mathcal{H}_{N}(\psi_{U^{c}}))\cdot\exp(3s_{N}N).$

For any $\psi\in\mathcal{U}$, we have a bound on the maximum outside of $U$,
and the gradient control. Thus, if we take

$\displaystyle\tilde{\mathcal{B}}_{j,U^{c}}:=\left\\{\psi_{U^{c}}\in\mathcal{B}_{j,U^{c}}:\left\|\psi_{U^{c}}\right\|_{\infty}\leqslant\sqrt{s_{N}N}\text{
and }\left\|\psi_{\partial U_{ext}}\right\|_{2}^{2}\leqslant 3s_{N}N\right\\}$
(53)

then it follows that

$\displaystyle\mathcal{U}\cap\mathcal{A}_{i,j,U}\subset\mathcal{B}_{i,U}\times\tilde{\mathcal{B}}_{j,U^{c}}.$
(54)

This completes the separation of the partition function. We have

$\displaystyle Z_{n}\leqslant\exp(3\theta s_{N}N)\cdot\sum_{|U|\leqslant
s_{N}^{-d-1}}\sum_{i+j\leqslant i^{*}}$
$\displaystyle\int_{\mathcal{B}_{i,U}}\exp(-\theta\mathcal{H}_{N}(\psi_{U}))\mathbf{d}\psi_{U}$
$\displaystyle\qquad\cdot\int_{\tilde{\mathcal{B}}_{j,U^{c}}}\exp(-\theta\mathcal{H}_{N}(\psi_{U^{c}}))\mathbf{d}\psi_{U^{c}}.$

######  Lemma 6.3.

Let $U$ be a fixed subset with size bounded above by $s_{N}^{-d-1}$, and
$\mathcal{B}_{i,U}$ be as in (50). We have

$\displaystyle\int_{\mathcal{B}_{i,U}}\exp(-\theta\mathcal{H}_{N}(\psi_{U}))\mathbf{d}\psi_{U}\leqslant\exp\left(-N\theta
I_{\nu}\left(\frac{i\kappa_{N}+\kappa_{N}}{N}\right)\right)\cdot\Theta_{1}(N)$

where

$\displaystyle\Theta_{1}(N)\leqslant{N^{s_{N}^{-d-1}}\pi^{s_{N}^{-d-1}}e^{C\kappa_{N}}}.$

###### Proof.

Since all sets $U$ under consideration have size bounded above $s_{N}^{-d-1}$
and the smallest non trivial cycle has size of order $N^{\frac{1}{d}}$, it
follows that for $N$ sufficiently large $U$ can be embedded in
$\mathds{Z}^{d}$. We assume that this is the case, fix an embedding, and as an
abuse of notation, refer to the corresponding images as $U$ and $\psi_{U}$. We
remind the reader that

$\displaystyle\mathcal{H}_{N}(\psi_{U})=\frac{N}{\nu}\cdot\mathcal{H}(N^{-1/2}\nu^{-1/2}\cdot\psi_{U}).$

We recall the function $I$ introduced in (11). By definition and using the
monotonicity of $I$, it follows that

$\displaystyle\exp(-\theta\mathcal{H}_{N}(\psi_{U}))\leqslant\exp\left(-\frac{N\theta}{\nu}I\left(\nu\cdot\frac{i\kappa_{N}+\kappa_{N}}{N}\right)\right).$

Thus,

$\displaystyle\int_{\mathcal{B}_{i,U}}\exp(-\theta\mathcal{H}_{N}(\psi_{U}))\mathbf{d}\psi_{U}\leqslant\exp\left(-\frac{N\theta}{\nu}I\left(\nu\cdot\frac{i\kappa_{N}+\kappa_{N}}{N}\right)\right)\cdot\text{Vol}(\mathcal{B}_{i,U}).$

The volume of $\mathcal{B}_{i,U}$ is easy to evaluate, as it is a concentric
shell with inner radius $\kappa$ and outer radius $\kappa i+\kappa$. Taking
the crude bound (the largest possible shell which has outer radius $N$) we
have

$\displaystyle\text{Vol}(\mathcal{B}_{i,U})\leqslant\frac{N^{|U|}\pi^{|U|}\kappa_{N}}{\Gamma(|U|+1)}\leqslant
N^{|U|}\pi^{|U|}\kappa_{n}.$

$\blacksquare$

Lemma 6.3 addresses the contribution to the free energy from the structured
portion of the Hamiltonian. We now establish the contribution from the
dispersive portion.

######  Lemma 6.4.

Let $U\subset V$ be a fixed subset and let $\tilde{\mathcal{B}}_{j,U^{c}}$ be
as defined in (53). We have

$\displaystyle\int_{\tilde{\mathcal{B}}_{j,U^{c}}}\exp(-\theta\mathcal{H}_{N}(\psi_{U^{c}}))\mathbf{d}\psi_{U^{c}}\leqslant\exp\biggl{(}N\log(\pi/\theta)-NW\left(\theta\cdot\frac{j\kappa_{N}+\kappa_{N}}{N}\right)\biggr{)}\cdot\Theta_{2}(N)\cdot\Theta_{2}^{\prime}(N,j)$

where

$\displaystyle\Theta_{2}(N)\leqslant\exp\left(10\theta s_{N}N\right)\text{ and
}\Theta^{\prime}_{2}(N,j)=\exp\left(2\kappa_{N}L\left(\theta\cdot\frac{j\kappa_{N}+\kappa_{N}}{N}\right)\right).$
(55)

###### Proof.

The $\ell^{\infty}$ bound on the $\psi\in\tilde{\mathcal{B}}_{j,U^{c}}$ tells
us that

$\displaystyle\|\psi_{U^{c}}\|_{p+1}^{p+1}\leqslant(s_{N}N)^{(p-1)/2}N.$

For the Hamiltonian, this yields

$\displaystyle\mathcal{H}_{N}(\psi_{U})=\left\|\nabla_{0}\psi_{U^{c}}\right\|_{2}^{2}-\frac{N^{(1-p)/2}}{p+1}\left\|\psi_{U^{c}}\right\|_{p+1}^{p+1}\geqslant\left\|\nabla_{0}\psi_{U^{c}}\right\|_{2}^{2}-\frac{1}{p+1}\cdot
N\cdot(s_{N})^{(p-1)/2}$

and in turn,

$\displaystyle\int_{\tilde{\mathcal{B}}_{j,U^{c}}}\exp(-\theta\mathcal{H}_{N}(\psi_{U^{c}}))\mathbf{d}\psi_{U^{c}}\leqslant\exp\bigl{(}\theta
Ns_{N}^{(p-1)/2}\bigr{)}\int_{\tilde{\mathcal{B}}_{j,U^{c}}}\exp(-\theta\left\|\nabla_{0}\psi_{U^{c}}\right\|)\mathbf{d}\psi_{U^{c}}.$
(56)

This enables us to discard the non linearity and consider only the free
portion of the Hamiltonian. The technique we will adopt is to, so to speak,
“patch” the missing portion to instead consider the free field on the torus.
It is easy to see, just by maximum bounds and calculating the volume of a box
with side length $\sqrt{3C_{d}\log|U|\cdot\theta^{-1}}$ that

$\displaystyle(6C_{d}\theta^{-1}\log
U)^{|U|}\cdot\exp\left(-6C_{d}d|U|\log|U|\right)\leqslant\int_{\\{\left\|\psi_{U}\right\|_{\infty}<\sqrt{3C_{d}\theta^{-1}\log|U|}\\}}\exp(-\theta\left\|\nabla_{0}\psi_{U}\right\|_{2}^{2})\mathbf{d}\psi_{U}.$

Thus (56) maybe bounded above by

$\displaystyle\left(\frac{\theta|U|^{2d}}{6C_{d}\log|U|}\right)^{|U|}\cdot\int_{\tilde{\mathcal{B}}_{j,U}\times\\{\left\|\psi_{U}\right\|_{\infty}\leqslant\sqrt{3C_{d}\log|U|}\\}}\exp\bigl{(}-\theta(\left\|\nabla_{0}\psi_{U}\right\|_{2}^{2}+\left\|\nabla_{0}\psi_{U^{c}}\right\|_{2}^{2})\bigr{)}\mathbf{d}\psi_{U^{c}}\cdot\mathbf{d}\psi_{U}$
(57)

Since we know that $\left\|\psi_{\partial U_{ext}}\right\|_{2}^{2}\leqslant
3s_{N}N$ and
$\left\|\psi_{U}\right\|_{\infty}\leqslant\sqrt{3C_{d}\theta^{-1}\log|U|}$,

$\displaystyle\sum_{{\boldsymbol{x}}\sim\mathbf{y}\in\partial
U}|\psi_{{\boldsymbol{x}}}-\psi_{\mathbf{y}}|^{2}\leqslant
6s_{N}N+6C_{d}\theta^{-1}|U|\log|U|.$

We have implicitly used the AM-GM inequality in the above bound. We obtain
that (57) may be further bounded above by

$\left(\frac{\theta|U|^{6C_{d}(1+d)}}{6C_{d}\log|U|}\right)^{|U|}\cdot\exp(Ns_{N}^{(p-1)/2})\cdot\exp(6Ns_{N})\int_{\tilde{\mathcal{B}}_{j,U^{c}}\times\\{\left\|\psi_{U^{c}}\right\|_{\infty}\leqslant\sqrt{3C_{d}\theta^{-1}\log|U|}\\}}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi$

Since $|U|\leqslant s_{N}^{-d-1}$, the largest mass that may be allocated to
$\psi_{U}$ is bounded above by $3\theta^{-1}C_{d}|U|\log|U|$. We may enlarge
our mass shell slightly so that for $N$ sufficiently large,

$\tilde{\mathcal{B}}_{j,U^{c}}\times\left\\{\left\|\psi_{U}\right\|_{\infty}\leqslant\sqrt{3C_{d}\theta^{-1}\log|U|}\right\\}\subset\mathcal{B}_{j,V}\cup\mathcal{B}_{j+1,V}.$

Combining all our bounds we have

$\displaystyle\int_{\tilde{\mathcal{B}}_{j,U^{c}}}\exp(-\theta\mathcal{H}_{N}(\psi_{U^{c}}))\mathbf{d}\psi_{U^{c}}\leqslant\Theta_{2}(N,U)\int_{\mathcal{B}_{j,V}\cup\mathcal{B}_{j+1,V}}\exp(-\theta\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi$

where

$\displaystyle\Theta_{2}(N,U):=\exp\biggl{(}\theta Ns_{N}^{(p-1)/2}$
$\displaystyle+6\theta Ns_{N}+\bigl{(}6C_{d}(1+d)\bigr{)}|U|\log|U|$
$\displaystyle+(\log\theta-\log 6C_{d})|U|-|U|\log\log|U|\biggr{)}.$

Using the bound $|U|\leqslant s_{N}^{-d-1}$, we keep only the leading order
terms in the error to obtain

$\Theta_{2}(N,U)\leqslant\left(8\theta Ns_{N}\right)$

To conclude, all that needs be done is apply Theorem 4.1, which tells us that

$\displaystyle\int_{\mathcal{B}_{j,V}\cup\mathcal{B}_{j+1,V}}\exp(-\theta\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi\leqslant\left(\frac{\pi}{\theta}\right)^{N}$
$\displaystyle\cdot\exp\left(-N\cdot
W\left(\theta\cdot\frac{j\kappa_{N}+\kappa_{N}}{N}\right)\right)$
$\displaystyle\cdot\exp\left(2\kappa_{N}L\left(\theta\cdot\frac{j\kappa_{N}+\kappa_{N}}{N}\right)+2\log
N\right).$

The $\log N$ addition to the error is far smaller than the leading term, we
may ignore it. With $\Theta^{\prime}_{2}(j)$ defined as in (55) , we conclude
the proof. $\blacksquare$

Having individually bounded the concentrated and dispersed contributions to
the partition function, we are now ready to combine them to yield the upper
bound of the limiting free energy.

###### Proof of Lemma 6.1.

We remind the reader that in the prior sections we have removed the explicit
$U$, and have all bounds in terms of the maximum allowed $|U|$. To avoid
cumbersome expressions, we begin by combining $\Theta_{1}$, $\Theta_{2}$ and
the errors arising from the separation at the boundary, carry out the sum over
$U$ to define

$\Theta_{S}:=\exp(3\theta
Ns_{N})\cdot\binom{N}{s_{N}^{-d-1}}\cdot\Theta_{1}(N)\cdot\Theta_{2}(N).$

Before proceeding further, we deal with the binomial coefficient. We have

$\binom{N}{s_{N}^{-d-1}}\leqslant e^{s_{N}^{-d-1}}\cdot N^{s_{N}^{-d-1}}.$
$\displaystyle Z_{N}\leqslant\Theta_{S}(N)\cdot\sum_{i+j\leqslant
i^{*}}\exp\left(\log\left(\frac{\pi}{\theta}\right)^{N}-\frac{N\theta}{\nu}\cdot
I\left(\nu\cdot\frac{i\kappa_{N}+\kappa_{N}}{N}\right)-N\cdot
W\left(\theta\cdot\frac{j\kappa_{N}+\kappa_{N}}{N}\right)\right)\cdot\Theta_{2}^{\prime}(j).$

Both $I$ and $W$ are monotonically decreasing. In particular, if
$j_{1}\leqslant j_{2}$, then

$\exp\left(-NW\left(\theta\cdot\frac{j_{1}\kappa_{N}+\kappa_{N}}{N}\right)\right)\leqslant\exp\left(-N\cdot
W\left(\theta\cdot\frac{j_{2}\kappa_{N}+\kappa_{N}}{N}\right)\right).$

For a given $i$, the largest value that $j$ can take is $i^{*}-i$. We use this
fact to also render the sum over $j$ irrelevant. We have

$\displaystyle
Z_{N}\leqslant\Theta_{S}\cdot\sum_{i=1}^{i^{*}}\Theta_{2}^{\prime}(i)\cdot\exp
N\left(\log\left(\frac{\pi}{\theta}\right)-\frac{\theta}{\nu}I\left(\nu\cdot\frac{i\kappa_{N}}{N}+\nu\cdot\frac{\kappa_{N}}{N}\right)-W\left(\theta\left(1-\frac{i\kappa_{N}}{N}\right)+2\theta\cdot\frac{\kappa_{N}}{N}\right)\right).$

In the above expression, we have used the fact that

$\frac{i^{*}\kappa_{N}}{N}-1\leqslant\frac{\kappa_{N}}{N},$

and have redefined $\Theta_{S}$ to include the additionally accrued $N$ from
summing over $j$. We now need to tackle the $i$ dependent error term
$\Theta_{2}^{\prime}$ and control the regularity of $W$. To do both, we bring
in Lemma 7.1. For $M>0$, we have an $i^{\prime}$ dependent on $M$ and $\theta$
such that for $i>i^{\prime}$,

$W\left(\theta\cdot\frac{(i^{*}-i)\kappa_{N}+\kappa_{N}}{N}\right)\geqslant
M+I(1).$

We also recall from Lemma 4.7 that

$L(b)\leqslant b^{-1},$

which tells us that

$\kappa_{n}L\left(\theta\frac{(i^{*}-i)\kappa_{N}+\kappa_{N}}{N}\right)\leqslant
N/\theta.$

Thus on choosing $M>1/\theta$,

$\displaystyle\sum_{i=i^{\prime}}^{i^{*}}\Theta_{2}^{\prime}(i)\cdot\exp
N\biggl{(}\log\bigl{(}\frac{\pi}{\theta}\bigr{)}-$
$\displaystyle\frac{\theta}{\nu}\cdot
I\bigl{(}\frac{i\kappa_{N}+\kappa_{N}}{N}\bigr{)}-W\bigl{(}\theta\cdot\frac{(i^{*}-i)\kappa_{N}+\kappa_{N}}{N}\bigr{)}\biggr{)}$
$\displaystyle\leqslant N\cdot e^{-N(M-1/\theta)}.$

When $i\leqslant i^{\prime}$, the same argument tells us that
$\Theta^{\prime}_{2}$ is uniformly bounded by $\exp(\kappa_{N}\cdot
C^{\prime})$ where $C^{\prime}$ is a constant depending on $\theta$, $\nu$ and
$M$. Further, in this case both $I$ (rescaled by $\nu$) and $W$ (rescaled by
$\theta$) are Lipschitz functions. Let the upper bound on both Lipschitz
constants be denoted by $C_{L}$ depending on $\theta$, $\nu$ and $M$. Taking
$i\kappa_{N}/N$ to be $a$, we discard the perturbations to the arguments of
the $I$ and $W$ functions obtaining

$Z_{N}\leqslant e^{C\kappa_{N}}\cdot\Theta_{S}\cdot\sum_{i=1}^{i^{\prime}}\exp
N\left(\log\left(\frac{\pi}{\theta}\right)-\theta\nu^{-1}I(\nu
a)-W(\theta(1-a))\right)+\Theta_{S}\cdot Ne^{-C_{M}N}$

We then obtain the trivial bound by optimizing over $a$ and finally rendering
the sum over $i$ irrelevant. We have

$Z_{N}\leqslant N\cdot e^{C\kappa_{N}}\cdot\Theta_{S}\cdot\exp
N\left(\log\left(\frac{\pi}{\theta}\right)-\inf_{a}\bigl{(}\theta\nu^{-1}I(\nu
a)-W(\theta(1-a))\bigr{)}\right)+\Theta_{S}\cdot e^{-C_{M}\cdot N}.$

We remind the reader that by Lemma 7.1 the constant $C_{M}$ may be chosen such
that

$C_{M}-\inf_{a}\left(\theta\nu^{-1}I(\nu a)+W(\theta(1-a))\right)\geqslant M.$

Thus,

$Z_{N}\leqslant
e^{C\kappa_{N}}\cdot\Theta_{S}\cdot\exp\left(\log\left(\frac{\pi}{\theta}\right)-\inf_{a}\bigl{(}\theta\nu^{-1}I(\nu
a)-W(\theta(1-a))\bigr{)}\right)\cdot\left(1+e^{-CN}\right).$

Redefining $\Theta_{S}$ one last time to include the additionally accrued
error term $e^{C\kappa_{N}}$, we are done. The error accrued from the
$1+e^{-CN}$ is insignificant compared to the leading order error term and is
therefore ignored.

$\blacksquare$

###  6.2. Lower Bound

In this section, we will verify the corresponding lower bound of the limiting
free energy. The restriction of the region of integration for the lower bound
as chosen in (51) should now be motivated by results of Section 6.1, as we saw
this was the region of dominant contribution.

######  Lemma 6.5.

We have a constant $C$ and sequences $\gamma_{N}$ and $\Theta_{I}(N)$
depending on $\nu$, $\theta$, $\rho$ and $\nu$ such that

$\frac{1}{N}\log
Z_{N}(\theta,\nu)\geqslant\log(\pi/\theta)-\inf_{0<a<1}(W(\theta(1-a))+\theta\nu^{-1}I(a\nu))+\frac{1}{N}\log\Theta_{I}(N)$

where

$\Theta_{I}(N)\geqslant\exp(-N\theta\gamma_{N}-C\cdot\kappa_{N})$

and

$\lim_{N\to\infty}\gamma_{N}=0.$

As with the upper bound we and partition the mass of a function into the
structured and dispersive part. The lower bound will be established by
restricting the integral to a specifically chosen subset of functions within
$\mathcal{B}_{i,U}\times\mathcal{B}{j,U^{c}}$. For the results to come, it
will be helpful for the sake of conciseness to define

$\displaystyle\rho_{i}:=\frac{2i\kappa_{N}+\kappa_{N}}{2N}.$ (58)

We will take $U$ to be a cube of side length $\lfloor\log N\rfloor$. We will
restrict the solitonic portion of our integral to be centered about a specific
minimizing sequence. Let $U^{o}$ denote the set of interior points of $U$, and
recall that $\varphi^{a,U^{o}}$ denotes the Dirichlet minimizer on $U^{o}$
with mass $a$. We define

$\displaystyle\mathcal{C}_{i,U}:=\left\\{\psi_{U}\in\mathds{C}^{U}:\left\|\psi_{U}-N^{1/2}\cdot\varphi^{\rho_{i},U^{o}}\right\|_{2}^{2}\leqslant
N^{-\delta}\right\\},$ (59)

where $\delta>0$ and fixed. As for the dispersive portion, we will define

$\displaystyle\mathcal{C}_{j,U^{c}}:=\left\\{\psi_{U^{c}}\in\mathcal{B}_{j,U^{c}}:\left\|\psi_{U^{c}}\right\|_{\infty}\leqslant
2\sqrt{3C_{d}\log N}\right\\}.$ (60)

It is clear that $C_{i,U}\subset\mathcal{B}_{i,U}$ and the analogous inclusion
for $C_{j,U^{c}}$ is obvious by definition. Note that any function sampled
from $\mathcal{B}_{j,U}$ places a maximum mass of $N^{-\delta}$ on the
boundary, and any function sampled from $\mathcal{B}_{j,U^{c}}$ may place a
maximum mass of $6C_{d}d\cdot(\log N)^{d}$ on the boundary. Combining these
facts yields the boundary estimate required for separating the soliton and
dispersive parts,

$\displaystyle\sum_{{\boldsymbol{x}}\sim\boldsymbol{y}\in\partial
U}|\psi_{{\boldsymbol{x}}}-\psi_{\boldsymbol{y}}|^{2}\leqslant\sum_{{\boldsymbol{x}}\in\partial{U}_{int}}|\psi_{{\boldsymbol{x}}}|^{2}+\sum_{\boldsymbol{y}\in\partial
U_{ext}}|\psi_{\boldsymbol{y}}|^{2}\leqslant 12dC_{d}(\log N)^{d}.$ (61)

Applying (51), (47) and (61) we have

$\displaystyle\begin{split}Z_{N}&\geqslant\exp\bigl{(}-12C_{d}d(\log
N)^{d}\bigr{)}\cdot\sum_{i=1}^{i_{*}-1}\int_{\mathcal{C}_{i,U}}\exp(-\theta\mathcal{H}_{N}(\psi_{U}))\mathbf{d}\psi_{U}\\\
&\qquad\qquad\qquad\cdot\int_{\mathcal{C}_{i_{*}-i,U^{c}}}\exp(-\theta\mathcal{H}_{N}(\psi))\mathbf{d}\psi_{U^{c}}.\end{split}$
(62)

######  Lemma 6.6.

Let $\mathcal{B}_{j,U}$ be as in (59). Then we have

$\int_{\mathcal{C}_{i,U}}\exp(-\theta\mathcal{H}_{N}(\psi_{U}))\mathbf{d}\psi_{U}\geqslant\exp\left(-N\theta\nu^{-1}\cdot
I(\nu\rho_{i})\right)\cdot\Theta_{3}(N,i)$

where

$\Theta_{3}(i,N)\geqslant\exp\left(-7dN^{1/2}-N\gamma_{i,N}\right)$

and

$\gamma_{i,N}:=\max_{\psi\in B_{i,U}}|\nu^{-1/2}\mathcal{H}(\nu^{1/2}\cdot
N^{-1/2}\cdot\psi)-\nu^{-1}I(\nu\rho_{i})|.$

###### Proof.

Our region of integration $\mathcal{C}_{i,U}$ is the ball of radius
$N^{-\delta/2}$ centered around $\varphi^{\rho_{i},U^{o}}$. We will verify
that this region is appropriately close in $\ell^{2}$ to the actual minimizer
$\phi$, and thus the energy $\mathcal{H}_{N}(\psi)$ is close to
$N\nu^{-1}I(\nu\rho_{j})$

$\int_{\mathcal{C}_{i,U}}\exp(-\theta\mathcal{H}_{N}(\psi_{U}))\mathbf{d}\psi_{U}=\exp(-\theta
N\nu^{-1}I\left(\nu\rho_{i}\right))\cdot\int_{\mathcal{B}_{i,U}}\exp(\theta(N\nu^{-1}I\left(\rho_{i}\right)-\mathcal{H}_{N}(\psi)))\mathbf{d}\psi_{U}$

Let $\psi\in\mathds{C}^{U}$ have unit mass and $t\in[0,N^{-\delta/2}]$. Using
the Cauchy-Schwarz inequality and the fact that the discrete Laplacian is
bounded, we have

$\left\|\nabla_{0}(N^{1/2}\cdot\varphi^{U}+t\psi)\right\|^{2}_{2}-\left\|N^{1/2}\cdot\nabla_{0}\varphi^{U}\right\|_{2}^{2}\leqslant
4dN^{-\delta/2}\cdot\sqrt{N\rho_{i}}+2dN^{-\delta}\leqslant
5dN^{(1-\delta)/2}.$

Further,

$\left\|N^{1/2}\cdot\varphi^{U}+t\psi\right\|_{p+1}^{p+1}-\left\|N^{1/2}\cdot\varphi^{U}\right\|_{p+1}^{p+1}\leqslant
N^{(p+1)/2}\left\|\psi\right\|_{p+1}\int_{0}^{t\cdot
N^{-1/2}}(\left\|\varphi^{U}\right\|_{p+1}+s\left\|\psi\right\|_{p+1})^{p}\text{d}s$
$\leqslant
N^{(1+p)/2}\left\|\psi\right\|_{p+1}\left\|\varphi^{U}\right\|_{p+1}^{p}\int_{0}^{t\cdot
N^{-1/2}}\left(1+s\frac{\left\|\psi\right\|_{p+1}}{\left\|\varphi^{U}\right\|_{p+1}}\right)^{p}\text{d}s.$

For any function in $\mathds{C}^{U}$ with mass $a$, we know that the lowest
possible $\ell^{p+1}$ norm is attained by the uniform function and is given by
$a^{1/2}\cdot|U|^{(1-p)/2(p+1)}$, and the largest possible is attained by
concentrating all the mass onto a single site and given by $a^{1/2}$. Thus, we
have a bounded constant depending only on $a$ such that

$\frac{1}{N^{(p-1)/2}}\left(\left\|N^{1/2}\cdot\varphi^{U}+t\psi\right\|_{p+1}^{p+1}-\left\|N^{1/2}\cdot\varphi^{U}\right\|_{p+1}^{p+1}\right)\leqslant
C\cdot N^{(-\delta+1)/2}.$

We then note that

$|\nu^{-1}\mathcal{H}_{N}(\psi)-N\nu^{-1}I(\nu\rho_{i})|\leqslant\max_{\psi\in\mathcal{B}_{j,U}}|\mathcal{H}_{N}(\psi)-NI(\nu\rho_{i})|=NC_{i}$

and note that $C_{i}$ is bounded and converges to $0$ as $n\to\infty$ as a
simple consequence of Lemma 5.2. As $\mathcal{C}_{i,U}$ is a ball of radius
$N^{-\delta}$ in $\mathds{C}^{U}$, we observe

$\text{Vol}(\mathcal{C}_{j,U})=\frac{\pi^{|\log N|^{d}}}{\Gamma(1+|\log
N|^{d})}\cdot N^{-2\delta|\log N|^{d}}.$

Defining

$\displaystyle\Theta_{3}(N):=\exp\left(-6dN^{(1-\delta)/2}-N\gamma_{i,N}\right)\cdot\frac{\pi^{|\log
N|^{d}}}{\Gamma(1+|\log N|^{d})}N^{-2\delta|\log N|^{d}}$ (63)

and then taking $\delta=0$ completes the proof. We clearly have

$\Theta_{3}(N,i)\geqslant\exp\left(-6dN^{1/2}-N\gamma_{i,N}-2d|\log
N|^{d}\log\log N\right)\geqslant\exp\left(-7dN^{1/2}-N\gamma_{i,N}\right)$

$\blacksquare$

######  Lemma 6.7.

Let $\mathcal{C}_{j,U_{c}}$ be as in (60). We have

$\int_{\mathcal{C}_{j,U^{c}}}\exp(-\theta\mathcal{H}_{N}(\psi_{U_{c}}))\mathbf{d}\psi_{U^{c}}\geqslant\exp\left(N\log\left(\frac{\pi}{\theta}\right)-N\cdot
W\left(\theta\rho_{j}\right)\right)\cdot\Theta_{4}(j,N)$

where

$\Theta_{4}(j,N)\geqslant(3C_{d}\log|\log N|)^{-2|\log N|^{d}}\cdot
e^{-2\kappa_{N}L(\theta\rho_{j})}.$

###### Proof.

Since we seek a lower bound, we may immediately discard the non linear part of
the Hamiltonian to obtain

$\int_{\mathcal{C}_{j,U^{c}}}\exp(-\theta\mathcal{H}_{N}(\psi_{U^{c}}))\mathbf{d}\psi_{U^{c}}\geqslant\int_{\mathcal{C}_{j,U^{c}}}\exp(-\theta\left\|\nabla_{0}\psi\right\|_{2}^{2})\mathbf{d}\psi_{U^{c}}.$

The next step, just as in the case of the upper bound, is to “patch” the free
field to the entire torus. Merely from the fact that the exponential of a
negative number is bounded above by 1, we have

$\int_{\\{\left\|\psi_{U}\right\|_{\infty}\leqslant\sqrt{3C_{d}\log
N}\\}}\exp(-\theta\left\|\nabla_{0}\psi\right\|_{2}^{2})\mathbf{d}\psi_{U}\leqslant\left({3C_{d}\log
N}\right)^{|\log N|^{d}}$

On the product space
$\mathcal{C}_{j,U^{c}}\times\\{\left\|\psi_{U}\right\|_{\infty}\leqslant\sqrt{3C_{d}\log
N}\\}$, by definition,

$\left\|\psi\right\|_{2}^{2}\leqslant(j+1)\kappa_{N}+3C_{d}|\log
N|^{d+1}\text{ and }\left\|\psi_{U^{c}}\right\|_{2}^{2}\geqslant j\kappa_{N}.$

If we define

$\widehat{\mathcal{B}}_{j}:=\left\\{\psi\in\mathds{C}^{V}:\left\|\psi\right\|^{2}_{2}\in\bigl{[}j\kappa_{N}+\frac{\kappa_{N}}{4},j\kappa_{n}+\frac{3\kappa_{N}}{4}\bigr{)},\left\|\psi\right\|_{\infty}\leqslant\sqrt{3C_{d}\log
N}\right\\},$

it is clear that for $N$ sufficiently large,

$\widehat{\mathcal{B}}_{j}\subset\mathcal{C}_{j,U^{c}}\times\\{\psi_{U}:\left\|\psi_{U}\right\|_{\infty}\leqslant\sqrt{3C_{d}\log
N}\\}.$

Thus,

$\displaystyle\int_{\mathcal{C}_{j,U^{c}}}\exp(-\theta\left\|\nabla_{0}\psi\right\|_{2}^{2})\mathbf{d}\psi_{U^{c}}\geqslant\frac{1}{(3C_{d}\log
N)^{|\log
N|^{d}}}\int_{\widehat{\mathcal{B}}_{j}}\exp(-\theta(\left\|\nabla_{0}\psi_{U}\right\|_{2}^{2}+\left\|\nabla_{0}\psi_{U^{c}}\right\|))\mathbf{d}\psi.$
(64)

The functions under consideration are bounded above, and we also know the size
of the boundary of $U$ is bounded above by $4d|\log N|^{d-1}$. This gives us
the gradient control required for patching. We have

$\sum_{{\boldsymbol{x}}\sim\boldsymbol{y}\in\partial
U}|\psi_{{\boldsymbol{x}}}-\psi_{\boldsymbol{y}}|^{2}\leqslant 36dC_{d}(\log
N)^{d}.$

We use this to further lower bound (64) as

$\frac{1}{(3C_{d}\log N)^{|\log N|^{d}}}\cdot\exp(-36dC_{d}(\log
N)^{d})\int_{\widehat{\mathcal{B}}_{j}}\exp(-\theta\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi.$

Using a combination of Theorems 4.1 and 4.11, we have that

$\displaystyle\int_{\mathcal{B}_{j}}\exp(-\theta\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi$
$\displaystyle\qquad\geqslant\left(\frac{\pi}{\theta}\right)^{N}\cdot\exp\bigl{(}-NW(\theta\rho_{j})\bigr{)}\cdot\frac{N(\theta\rho_{i}-\theta
C_{d})_{+}+\kappa_{N}}{N\theta\rho_{i}+\kappa_{N}}\cdot
e^{-2\kappa_{N}L(\theta\rho_{j})}\cdot\left(1-\frac{2cm_{N}(2)N}{\kappa_{N}^{2}}\right)^{2}.$

Define

$\Theta_{4}:=\frac{\exp(-36dC_{d}(\log N)^{d})}{(3dC_{d}\log N)^{|\log
N|^{d}}}\cdot\frac{\kappa_{N}}{N+\kappa_{N}}\cdot\left(1-\frac{2cm_{N}(2)N}{\kappa_{N}^{2}}\right)^{2}e^{-2\kappa_{N}L(\theta\rho_{j})}$

and observe that

$\Theta_{4}(N,j)\geqslant\exp\left(-4\kappa_{N}L(\theta\rho_{j})\right)\cdot\left(1-\frac{2cm_{N}(2)N}{\kappa_{N}^{2}}\right)^{2}$

for $N$ sufficiently large. $\blacksquare$

Again, having bounded the solitonic and free contributions from a given shell
below, we are now ready to establish the free energy lower bound. Compared to
the upper bound, the process is much easier, as it only involves restricting
to the appropriate mass shell.

###### Proof of Lemma 6.5.

We remind the reader that we are taking $j=i_{*}-1$. All we need to do is
restrict the sum (62) to the $i$ that yields $\rho_{i}$ closest to the value
of $a_{\star}$. Let $i_{n}$ be a sequence such that $\rho_{i_{n}}\to
a_{\star}$ as $n\to\infty$. Note, by the definition of $i_{*}$, we have

$\rho_{j}=\frac{2j\kappa_{N}+\kappa_{N}}{2N}=\frac{2i_{*}\kappa_{N}-2i\kappa_{N}+\kappa_{N}}{N}=\frac{i_{*}\kappa_{N}+\kappa_{N}}{N}-\rho_{i}\leqslant
1-\rho_{i}.$

Combining,

$Z_{N}\geqslant\exp(-12dC_{d}(\log
N)^{d})\cdot\Theta_{3}(N,i)\cdot\Theta_{4}(N,i_{*}-i)\cdot\left(\frac{\pi}{\theta}\right)^{N}\cdot\exp\biggl{(}-N\nu^{-1}\theta
I(\nu\rho_{i})-NW(\theta(1-\rho_{i}))\biggr{)}.$

Now note that $|i_{n}-a_{*}|\leqslant\kappa/N$, and we know that for a fixed
$M>0$, $a_{*}\leqslant a_{M}<1$ by Lemma 7.1. This tells us two things,
firstly that $\Theta_{4}$ may be removed of its $j$ dependence, and secondly
on the interval $[0,a_{M}]$ the function

$W(\theta(1-a))+\theta\nu^{-1}I(\nu a)$

is Lipschitz, with Lipschitz constant depending only on $p$, $\theta$ and
$\nu$. Let the Lipschitz constant be $C_{2}$. Defining

$\Theta_{I}(n)=\Theta_{3}(n)\cdot\Theta_{4}(n)\cdot\exp(-4d\tau(\log
n)^{d})\cdot\exp(-(C+C_{2})\kappa_{N})$

completes the proof. Clearly, we must have appropriately chosen constant

$\Theta_{I}(N)\geqslant\exp\left(-C\kappa_{N}-N\gamma_{N}\right)\cdot\left(1-\frac{2cm_{N}(2)N}{\kappa_{N}^{2}}\right)^{2}$

where now $\gamma_{N}$ is defined for the optimizing $a_{\star}$.
$\blacksquare$

Thus, we have assembled all the requisite ingredients for Theorem 2.3.

##  7\. Analysis of Phases

Recall that the optimizing mass fractions $a_{\star}$ characterize the phase
behavior. The optimizing values $a_{\star}$ tell us the proportions of the
mass that are energetically favourable to allocate to the soliton
contribution. We begin this section with the following effective bound on
$a_{\star}$, which was used crucially in proving convergence of free energy.
Further, this establishes that it is never energetically favourable to have
all mass allocated to the soliton. Recall the expression for the limiting free
energy

$F(\theta,\nu)=\log\frac{\pi}{\theta}-\inf_{0<a<1}\left(W(\theta(1-a))+\frac{\theta}{\nu}I(a\nu)\right).$

######  Lemma 7.1.

We have $\mathscr{M}(\theta,\nu)\subseteq[0,a_{M}]$ where

$a_{M}:=1-\exp(-\theta(2d-I(\nu)/\nu)).$

###### Proof.

Using the fact that $J(r)=I(r)/r$ is decreasing and properties of
$\widehat{W}$ from Lemma 4.7, we get that

$\displaystyle W(\theta(1-a))-W(\theta)+\theta I(a\nu)/\nu$
$\displaystyle=\widehat{W}(\theta(1-a))-\widehat{W}(\theta)-\ln(1-a)+a\theta
J(a\nu)$ $\displaystyle\geqslant-2da\theta-\ln(1-a)+a\theta J(\nu)$
$\displaystyle=-\ln(1-a)-a\theta(2d-J(\nu))>0$

if $a>1-\exp(-\theta(2d-J(\nu)))$. In particular, we have

$\inf_{a>1-\exp(-\theta(2d-I(\nu)/\nu))}\left(W(\theta(1-a))+\theta
I(a\nu)/\nu\right)>W(\theta).$

This completes the proof. $\blacksquare$

It is clear that $\mathscr{M}$ is compact and closed in $[0,1)$ by the
continuity of

$a\mapsto G_{\theta,\nu}(a):=W(\theta(1-a))+\theta I(a\nu)/\nu.$

We begin with some simple results to show that there are regions each of the
phases are achieved. The easiest technique is to note that there are intervals
where $W$ and $I$ are constant. We define

$a_{W}(\theta):=1-\frac{C_{d}}{\theta}\text{ and
}a_{I}(\nu):=\frac{R_{p}}{\nu}.$

Clearly, $W(\theta(1-a))=C_{d}$ for all $a\leqslant a_{W}$ and $I_{\nu}(a)=0$
for all $a\leqslant a_{I}$.

######  Lemma 7.2 (Phase 1).

Let $\nu\leqslant R_{p}$. Then $0\in\mathcal{F}$.

###### Proof.

Since $v\leqslant R_{p}$, then $a_{I}\geqslant 1$, and thus $I_{\nu}(a)=0$ for
all values of $a$. It follows that $G_{\theta,\nu}$ is non decreasing in $a$,
and thus $0\in\mathcal{F}$. $\blacksquare$

This lemma shows that the dispersive phase is indeed achieved. We show that
the soliton phase is achieved as well.

######  Lemma 7.3 (Phase 2).

Let $\nu$ and $\theta$ be such that
$\frac{C_{d}}{\theta}+\frac{R_{p}}{\nu}<1.$ Then
$0\notin\mathscr{M}(\theta,\nu)$.

###### Proof.

With $\nu$ and $\theta$ as above, we know that $a_{I}<a_{W}$. On the interval
$[0,a_{W}]$, the function $G_{\theta,\nu}$ is non-increasing and is in fact
strictly decreasing on the interval $[a_{I},a_{W}]$. Thus, there is an
$a\in[a_{I},a_{W}]$ such that $G_{\theta,\nu}(a)<W(\theta)=G_{\theta,\nu}(0)$.
$\blacksquare$

###  7.1. Existence of Transition Curve

We have shown that each of the defined phases may be realized, it remains to
be shown that the regions are separated by a curve. We also need to establish
the continuity of the curve.

######  Lemma 7.4.

Let $(\theta,\nu)$ be such that $0\in\mathscr{M}(\theta,\nu)$. We say
$(\theta_{1},\nu_{1})\preccurlyeq(\theta,\nu)$ if $\theta_{1}\leqslant\theta$
and $\nu_{1}\leqslant\nu$. Now let

$(\theta_{1},\nu_{1})\preccurlyeq(\theta,\nu)\preccurlyeq(\theta_{2},\nu_{2}).$

If $0\in\mathscr{M}(\theta,\nu)$, then $0\in\mathscr{M}(\theta_{1},\nu_{1})$.
Conversely, if $0\notin\mathscr{M}(\theta,\nu)$, then
$0\notin\mathscr{M}(\theta_{2},\nu_{2})$.

###### Proof.

We know that $0\in\mathscr{M}(\theta,\nu)$ if and only if for all $a\in[0,1)$,

$\displaystyle W(\theta)\leqslant W(\theta(1-a))+\theta\nu^{-1}I(\nu a).$ (65)

Using the fact that $W^{\prime}(b)=L(b)$, we may rewrite (65)

$\displaystyle I_{\nu}(a)+\int_{(1-a)}^{1}L(\theta s)ds\geqslant 0$ (66)

By Lemma 5.9, decreasing $\nu$ increases $I_{\nu}(a)$, preserving the
inequality. In addition, decreasing $\theta$ increases $L(\theta s)$, still
preserving the inequality. As for the converse, observe that
$0\notin\mathscr{M}$ if and only if we have an $a_{1}$ such that

$\displaystyle I_{\nu}(a_{1})+\int_{(1-a_{1})}^{1}L(\theta s)ds<0.$ (67)

Again, by Lemma 5.9, increasing $\nu$ decreases $I_{\nu}(a)$, and increasing
$\theta$ decreases $L(\theta s)$. $\blacksquare$

Lemma 7.4 is adequate to establish not only the existence of the transition
curve but the monotonicity as well. We remark now the following useful
consequence of the relation (67), the set of $(\theta,\nu)$ corresponding to
the soliton phase is open in the plane $[R_{p},\infty)\times[0,\infty)$.

######  Corollary 7.5.

Let $\theta_{c}:(R_{p},\infty)\to(0,\infty)$ be the measurable function
defined as

$\theta_{c}(\nu):=\sup\\{\theta>0:0\in\mathcal{F}(\theta,\nu)\\}.$

By definition, $\theta_{c}$ is our transition curve. We have that $\theta_{c}$
is non-increasing and continuous.

###### Proof.

By Lemma 7.3, we know that

$\theta_{c}(\nu)\leqslant\frac{C_{d}\nu}{\nu-R_{p}}<\infty$

for all $\nu$ under consideration. Not only does this tell us that
$\theta_{c}$ is finite, it also tells us that it may be evaluated at every
$\nu$ under consideration. Next, suppose we have that $\nu_{1}<\nu_{2}$ such
that $\theta_{c}(\nu_{1})<\theta_{c}(\nu_{2})$. Observe that this contradicts
Lemma 7.4. Since we have proved that $\theta_{c}$ is non-increasing, it
further implies that on any compact interval
$[\nu_{1},\nu_{2}]\subset(R_{p},\infty)$ there are at most countably many
discontinuities. Let $\nu_{3}$ be a point of discontinuity. We have that

$\theta_{1}:=\lim_{\nu\downarrow\nu_{3}}\theta_{c}(\nu)\leqslant\theta_{2}:=\theta_{c}(\nu_{c})\leqslant\theta_{3}:=\lim_{\nu\uparrow\nu_{3}}\theta_{c}(\nu),$

Suppose

$\theta_{2}<\theta_{3}.$

We may find an $\varepsilon>0$ such that $\theta_{2}+\varepsilon<\theta_{3}$.
By hypothesis,

$0\notin\mathcal{F}(\nu_{3},\theta_{2}+\varepsilon).$

However, note that every open ball centered at
$(\nu_{3},\theta_{2}+\varepsilon)$ intersects
$[R_{p},\nu_{3})\times[0,\theta_{3})$ on which $0\in\mathcal{F}(\theta,\nu)$,
which contradicts the fact that the solitonic region is open in the plane, and
we must have $\theta_{2}=\theta_{3}$. To actually establish continuity, we
note that the boundary between the solitonic region and the plane may be
characterized as all $(\theta,\nu)$ such that (65) holds, and we have an
$a_{\star}\geqslant a_{I}$ such that

$I_{\nu}(a_{\star})+\int_{(1-a\star)}^{1}L(\theta s)ds=0.$

Let $\varepsilon>0$ be such that $\theta_{1}+2\varepsilon<\theta_{2}$. We have
that $(\nu,\theta_{1}+\varepsilon)$ is on the boundary, as for any
$\nu^{\prime}>\nu$ we enter the soliton phase. Further, we know that

$\theta_{1}+\varepsilon<\theta_{2}\leqslant\frac{C_{d}\nu}{\nu-R_{p}}.$

Thus, $a_{\star}>a_{W}$ and

$\int_{(1-a_{\star})}^{1}L((\theta_{1}+\varepsilon)s)ds>\int_{(1-a_{\star})}^{1}L((\theta_{1}+1.5\varepsilon)s)ds.$

Combined with the fact that $I_{\nu}(a_{\star})<0$, this tells us that
$(\nu,\theta_{1}+1.5\varepsilon)$ is in the soliton phase, further implying
that $\theta_{2}\leqslant\theta_{1}+\varepsilon$ which is a contradiction.
Thus, we must have that $\theta_{1}=\theta_{2}$, which verifies continuity.
$\blacksquare$

######  Remark 7.6.

The phase curve could be equivalently defined with $\nu_{c}$ as a function of
$\theta$, which would be the exact inverse of the function $\theta_{c}$
obtained above. The same properties of monotonicity and continuity can be
verified for $\nu_{c}$, which then tells us that $\theta_{c}$ must be strictly
decreasing.

##  8\. Discussion on the Typical Function in the Subcritical Phase

In this section, we will work with the condition that $1<p<5+4\sqrt{2}$, and
establish some properties of a typical function sampled from the measure
$\mu_{N}$ in the dispersive phase. Recall that this corresponds to
$(\theta,\nu)\in\mathcal{D}$. We will also further make the assumption that
$\theta<C_{d}$, for reasons that will be made clear. In this section, it will
be convenient to work with the following rescaled version of the measure
$\mu_{N}$, which is equivalent due to Lemma 3.1.

$\tilde{\mu}_{N}(\mathcal{A}):=\frac{1}{\tilde{Z}(\theta,\nu)}\int_{\mathcal{A}}\exp\left(-\left\|\nabla\psi\right\|^{2}_{2}+(\theta
N)^{-\frac{p-1}{2}}\frac{2\nu}{p+1}\left\|\psi\right\|_{p+1}^{p+1}\right)\mathds{1}_{\left\|\psi\right\|^{2}_{2}\leqslant\theta
N}\cdot\mathbf{d}\psi$

We will define the reference measure $\mu_{\text{ref}}$, and will then regard
$\tilde{\mu}_{N}$ as an exponential tilt with respect to the nonlinearity. We
define

$\mu^{\theta}_{\text{ref}}(\mathcal{A}):=\frac{1}{Z_{\text{ref}}(\theta)}\int_{\mathcal{A}}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi.$

We begin by explicitly demonstrating the sense in which this measure is close
to the massive Gaussian free field.

######  Theorem 8.1.

Let $y_{N}$ be defined by the equation $K^{\prime}_{N}(y_{N})=\theta$, and let
$Z^{y_{N}}(\theta)$ denote the partition function of the massive free field on
the torus, given by

$Z^{y_{N}}=\frac{\pi^{N}}{\det(y_{N}-\Delta)}.$

Then we have a constant $C$ depending on $\theta$ such that

$\lim_{N\to\infty}\frac{e^{-N\theta}\cdot
Z^{y_{N}}}{\sqrt{N}Z_{\text{ref}}(\theta)}=C.$

###### Proof.

We have that

$\displaystyle 1$
$\displaystyle=\frac{1}{Z_{\text{ref}}(\theta)}\int_{\left\|\psi\right\|_{2}^{2}\leqslant\theta
N}\exp(-\left\|\nabla\psi\right\|_{2}^{2})\mathbf{d}\psi$
$\displaystyle=\frac{e^{N\theta}\cdot
Z^{y_{N}}}{Z_{\text{ref}}(\theta)}\operatorname{\mathds{E}}\exp\left(\left\|\Psi^{y_{N}}\right\|_{2}^{2}-N\theta\right)\cdot\mathds{1}_{\left\|\Psi^{y_{N}}\right\|_{2}^{2}\leqslant
N\theta}$

It, therefore, suffices to prove that