# Strong Characterization for the Airy Line Ensemble

Amol Aggarwal1,2 and  Jiaoyang Huang3 Department of Mathematics, Columbia
University  Clay Mathematics Institute  Department of Statistics and Data
Science, University of Pennsylvania

###### Abstract.

In this paper we show that a Brownian Gibbsian line ensemble whose top curve
approximates a parabola must be given by the parabolic Airy line ensemble.
More specifically, we prove that if
$\bm{\mathcal{L}}=(\mathcal{L}_{1},\mathcal{L}_{2},\ldots)$ is a line ensemble
satisfying the Brownian Gibbs property, such that for any $\varepsilon>0$
there exists a constant $\mathfrak{K}(\varepsilon)>0$ with

$\displaystyle\mathbb{P}\Big{[}\big{|}\mathcal{L}_{1}(t)+2^{-1/2}t^{2}\big{|}\leq\varepsilon
t^{2}+\mathfrak{K}(\varepsilon)\Big{]}\geq 1-\varepsilon,\qquad\text{for all
$t\in\mathbb{R}$},$

then $\bm{\mathcal{L}}$ is the parabolic Airy line ensemble, up to an
independent affine shift. Specializing this result to the case when
$\bm{\mathcal{L}}(t)+2^{-1/2}t^{2}$ is translation-invariant confirms a
prediction of Okounkov and Sheffield from 2006 and Corwin–Hammond from 2014.

###### Contents

  1. 1 Results and Preliminaries
    1. 1 Introduction
    2. 2 Results
    3. 3 Proof of Characterization
    4. 4 Miscellaneous Preliminaries
  2. 2 Gap Monotonicity and Likelihood of On-Scale Events
    1. 5 Gap Monotonicity
    2. 6 Likelihood of Medium Position Events
    3. 7 Likelihood of On-Scale and Improved Medium Events
  3. 3 Airy Statistics From Non-Intersecting Brownian Bridges
    1. 8 Gap Convergence to the Airy Point Process
    2. 9 Airy Line Ensembles From Airy Point Processes
    3. 10 Limit Shapes for Non-intersecting Brownian Bridges
    4. 11 Second Derivative Approximations for Paths
    5. 12 Proofs of Results From Section 11.3
  4. 4 Limit Shapes Near the Edge
    1. 13 Density Estimates for Limit Shapes
    2. 14 Limit Shapes on Tall Rectangles
  5. 5 Couplings on Tall Rectangles
    1. 15 Concentration Bounds and Extreme Path Estimates
    2. 16 Boundary Removal Coupling
    3. 17 Existence of Preliminary Couplings
    4. 18 Improved Hölder Estimates
  6. 6 Global Law and Regular Profiles
    1. 19 Likelihood of Regular Profile Events
    2. 20 Proof of the Global Law
  7. 7 Appendices
    1. 21 Proofs of Results From Chapter 1
    2. 22 Proofs of Results From Chapter 2
    3. 23 Proofs of Results From Chapter 3
    4. 24 Proofs of Results From Chapter 4
    5. 25 Convergence of the KPZ and Log-Gamma Line Ensembles

##  Chapter 1 Results and Preliminaries

###  1\. Introduction

####  1.1. Preface

A fundamental question in probability theory and mathematical physics concerns
the classification of Gibbs measures for statistical mechanical systems. In
this paper we analyze such questions for _Brownian Gibbsian line ensembles_ ,
which are infinite sequences of random functions (or curves)
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$, with each
$\mathsf{x}_{j}:\mathbb{R}\rightarrow\mathbb{R}$ continuous, that satisfy the
_Brownian Gibbs property_. The latter imposes two constraints. The first is
that the $\mathsf{x}_{j}$ are ordered, meaning that
$\mathsf{x}_{1}>\mathsf{x}_{2}>\cdots$ almost surely. The second is a
resampling condition indicating that $\bm{\mathsf{x}}$ behaves as a family of
two-sided Brownian motions conditioned to never intersect. More specifically,
for any integers $1\leq i\leq j$ and real numbers $a<b$, upon conditioning on
$\mathsf{x}_{k}(s)$ with either $k\notin[i,j]$ or $s\notin[a,b]$, the law of
the remaining $(\mathsf{x}_{i},\mathsf{x}_{i+1},\ldots,\mathsf{x}_{j})$ on
$[a,b]$ is given by standard Brownian bridges (whose starting and ending
points are determined by the conditioning) conditioned to not intersect, stay
below $\mathsf{x}_{i-1}$, and stay above $\mathsf{x}_{j+1}$ (that is, to
satisfy $\mathsf{x}_{i-1}>\mathsf{x}_{i}>\cdots>\mathsf{x}_{j+1}$, where
$\mathsf{x}_{0}=\infty$). See Figure 1.1 for a depiciton.

A prominent example of a Brownian Gibbsian line ensemble is the _parabolic
Airy line ensemble_ , introduced by Prahöfer–Spohn [107] as the scaling limit
for the multi-layer polynuclear growth (PNG) model; it can also be viewed as
the edge limit for $n$ non-intersecting Brownian bridges, sometimes called the
Brownian watermelon. These models are exactly solvable, or integrable, through
the framework of determinantal point processes. In [107], and the subsequent
work of Johansson [81], the multi-point correlation functions of the Airy line
ensemble were computed in terms of Airy functions. These calculations in
particular implied that its curves decay parabolically, but become jointly
translation-invariant after simultaneously shifting them by a parabola. The
top curve of the Airy line ensemble is known as the Airy2 process, whose one-
point marginal is the Tracy–Widom disitribution governing fluctuations for the
largest eigenvalue of a Gaussian Unitary Ensemble (GUE) random matrix [119].
By combining these integrable inputs with a probabilistic analysis,
Corwin–Hammond [34] realized the parabolic Airy line ensemble as a family of
continuous functions satisfying the Brownian Gibbs property (that is, as a
Brownian Gibbsian line ensemble); the Airy line ensemble (incorporating the
above parabolic shift) was later shown by Corwin–Sun [39] to be ergodic under
translations.

Over the past two decades, the Airy line ensemble has become a central object
in random surfaces and stochastic growth models. In particular, it has long
been understood that many random surfaces exhibit boundary-induced phase
transitions, in that they can admit sharp interfaces separating faceted
regions (where the surface is almost deterministically flat) from rough ones
(where it appears more random). For Ising crystals, this phenomenon dates back
to the Wulff construction (see the books of Dobrushin–Kotecký–Shlosman [52]
and Cerf [29]) and, for other surfaces, to the work of Jockusch–Propp–Shor
[77] (who studied random domino tilings of the Aztec diamond).

Figure 1.1. Depicted above is an example of Brownian Gibbsian line ensemble,
where the red curves can be resampled in the shaded region.

At this interface, also called the arctic boundary or facet edge, the level
lines of the random surface height function are believed to exhibit $n^{1/3}$
fluctuations on domains of diameter $n$; this $1/3$ exponent is closely
related to the Pokrovsky–Talapov law that predicts the behavior of facet
transitions in two-dimensional crystals [105, 106]. Upon rescaling by
$n^{1/3}$, it is further believed that these level lines converge to the Airy
line ensemble in the large $n$ limit. While this prediction remains unproven
in general, it has been established for various solvable models, starting with
the Brownian watermelon in [81] and random plane partitions by
Okounkov–Reshetikhin in [102, 103]. We refer to the survey of Johansson [84]
for an exposition and extensive list of further references, as well as to the
work of Ferrari–Shlosman [58] for additional predictions in this direction.

The relation to stochastic growth models is that their height fluctuations
(under wedge initial data) should converge to the Airy2 process, a ubiquitous
feature of systems in the Kardar–Parisi–Zhang (KPZ) universality class [88];
see the surveys of Corwin [31] and Quastel [108]. One explanation for this is
that, at least in some cases, these models can be exactly mapped to the facet
edge of a corresponding random surface, also sometimes called a Gibbsian line
ensemble (as the level lines of the surface height function form a line
ensemble satisfying a Gibbs property). This idea was initially applied in
[77], which used the shuffling algorithm introduced by
Elkies–Kuperberg–Larsen–Propp [54] to map the discrete-time totally asymmetric
simple exclusion process (TASEP) to the arctic boundary for a random domino
tiling. Following the framework of Rost [110], [77] showed a hydrodynamical
limit for this TASEP, yielding the limit shape for the arctic boundary.

Such correspondences have since been more fruitfully used in reverse, to show
that the Airy fluctuations for random surfaces imply those for stochastic
growth models. This was first applied to analyze determinantal systems, such
as TASEPs [78, 82] through random tilings, and PNG models [80, 107, 81]
(generalizations of longest increasing subsequences of random permutations,
studied by Baik–Deift–Johansson [12]) and Brownian last passage percolation
(by Baryshnikov [15], O’Connell–Yor [101, 98], and Warren [121]) through
Brownian watermelons. See [118, 83, 59] for surveys on these earlier papers.
Later work of Hammond [74, 71, 72, 73] used the associated line ensembles to
provide a detailed probabilistic analysis of the on-scale polymer geometry for
Brownian last passage percolation. More recent papers of
Matetski–Quastel–Remenik [93] and Dauvergne–Ortmann–Virág [44] analyzed the
full space-time scaling limit for TASEPs and last passage models, under
arbitrary initial data. The latter in particular showed how the above
correspondences with random surfaces (for them, the Brownian watermelon) could
be used to describe this limit entirely in terms of the Airy line ensemble,
further solidifying its role in the KPZ universality class. There is now a
vast literature utilizing Gibbsian line ensembles to elucidate the
probabilistic structure behind KPZ models. For examples just in the last
several years, we refer to the papers (and references therein) [27, 104, 111,
43] that used line ensembles to prove Brownian comparison results for KPZ
models; [33, 40, 62, 122] that used them to analyze the fine structure of the
continuum directed polymer; [16, 46, 61, 63, 41] that used them to examine the
fractal behavior of the directed landscape; and [60, 36, 42] that used them to
study exceptional times, and related applications, for the KPZ fixed point.

The reasons for the effectiveness of random surface models, in understanding
convergence to Airy statistics, can be viewed as twofold. The first reason is
algebraic; if the model is integrable, then its solvable underpinnings often
become more visible when one examines the random surface as a whole, as
opposed to only its arctic boundary. Indeed, the former combinatorially
corresponds to a Gelfand–Tsetlin pattern, which contains significantly more
structure than the latter, which corresponds to its first (or last) column.
This structure enables the introduction of natural $2+1$ dimensional dynamics
on these random surface models, which project precisely to many of the $1+1$
dimensional growth systems in the KPZ universality class. See the works of
Borodin–Ferrari [23], O’Connell [99], and Borodin–Corwin [21] for examples of
this perspective.

The second (which is more relevant to the impetus of this paper) is
probabilistic and relates to the Gibbs property satisfied by random surfaces
defined by local Boltzmann weights. Although the microscopic Gibbs property
behind such a model might depend on the details of its definition, the general
intuition is that this Gibbs property should converge to the Brownian one
around a facet edge. Indeed, in such regions, the random surface becomes more
flat, so its level lines become more sparse and separated. Hence, any local
interactions between them should be asymptotically lost, making these level
lines behave as random walks that do not intersect. Taking their scaling
limit, one then expects to find an infinite family of non-intersecting
Brownian bridges.

Facilitated by the extensive array of methodology to show convergence to
Brownian bridges, the above heuristic has been justified for wide classes of
random surfaces, both solvable and not. In particular, assuming certain
tightness and curvature conditions for their topmost curve, it has been proven
that any limit point for the edge of such Gibbsian line ensembles must be
Brownian ones; see the works of Dimitrov–Wu [51, 50],
Barraquand–Corwin–Dimitrov [14], and Serio [114]. Ideas of this nature had
earlier been used to prove qualitative results (such as local Brownian
continuity for the height function) for integrable, but non-determinantal,
models in the KPZ universality class. These include the KPZ equation and
O’Connell–Yor polymer [35]; asymmetric simple exclusion process and stochastic
six-vertex model [32]; and log-gamma polymer [123, 14] (where the more
involved associated Gibbsian line ensembles arose from works of
O’Connell–Warren [100] and Nica [97]; Borodin–Bufetov–Wheeler [20]; and
Corwin–O’Connell–Seppäläinen–Zygouras [37] and Johnston–O’Connell [85],
respectively).

The above frameworks are well-suited to the qualitative task of showing that
any edge limit for a random surface must be a Brownian Gibbsian line ensemble
(up to tightness and curvature constraints for the extreme level line).
However, they do not address the quantitative task of pinning the limit down
as the parabolic Airy line ensemble. Therefore, a basic question that arises
is if there is an axiomatic characterization of the Airy line ensemble or,
more specifically, some practical criterion for when a Brownian Gibbsian line
ensemble must be the parabolic Airy one. The purpose of this paper is to
establish such a criterion.

This criterion can be stated as follows (see 2.8 below). Let
$\bm{\mathcal{L}}=(\mathcal{L}_{1},\mathcal{L}_{2},\ldots)$ denote a Brownian
Gibbsian line ensemble. Suppose for any $\varepsilon>0$ that there is a
constant $\mathfrak{K}(\varepsilon)>0$ with

(1.1)
$\displaystyle\mathbb{P}\Big{[}\big{|}\mathcal{L}_{1}(t)+2^{-1/2}t^{2}\big{|}\leq\varepsilon
t^{2}+\mathfrak{K}(\varepsilon)\Big{]}\geq 1-\varepsilon,\qquad\text{for each
$t\in\mathbb{R}$}.$

Then $\bm{\mathcal{L}}$ is the parabolic Airy line ensemble, up to an
independent affine shift; see 2.9 below.

Informally, (1.1) states that the top curve $\mathcal{L}_{1}$ of
$\bm{\mathcal{L}}$ likely satisfies
$\mathcal{L}_{1}(t)=-\big{(}2^{-1/2}+o(1)\big{)}t^{2}$ (with the constant
$\mathfrak{K}(\varepsilon)$ in (1.1) being used to correct this approximation
for small $t$). Let us mention that some type of quadratic decay for
$\mathcal{L}_{1}$ must be imposed for the above characterization to hold. For
instance, the Airy line ensembles with wanderers introduced by
Adler–Ferrari–van Moerbeke [1] form examples of Brownian Gibbsian line
ensembles for which $\mathcal{L}_{1}$ only decays linearly.

Observe that (1.1) incorporates the scenario when the parabolically shifted
line ensemble $\bm{\mathcal{L}}(t)+2^{-1/2}t^{2}$ is translation-invariant in
$t$. In this case, it was predicted111It was unpublished at the time but has
since appeared in various forms in print, such as [34, Conjecture 3.2] and
[39, Conjecture 1.7]. by Okounkov and Sheffield in 2006 that
$\bm{\mathcal{L}}$ is given by a parabolic Airy line ensemble, up to an
independent overall constant shift. This is in the spirit of classifications
for translation-invariant Gibbs measures of discrete random surfaces, proven
by Sheffield [116] (but is also of a distinct nature, since here the base
space of the line ensemble is not discrete, and also since here translation-
invariance holds in only one coordinate, not both).

Our result 2.9 quickly implies this prediction (see 2.12 below), and further
generalizes upon it in two ways. First, our assumption (1.1) only constrains
the top curve of the ensemble, instead of imposing that all of its curves be
jointly translation-invariant. The notion that sufficient information on the
top curve could determine the entire line ensemble also appeared in the work
of Dimitrov–Matetski [49, 48], though the control they required was quite
siginificant, namely, knowledge of its full law (all of its finite-dimensional
marginals). Those results in particular implied that $\bm{\mathcal{L}}$ is a
parabolic Airy line ensemble, if one happened to know in advance that
$\mathcal{L}_{1}$ were an Airy2 process. Prior to our work, the latter seemed
to be quite an involved task, though had been done by Quastel–Sarkar [109] and
Virág [120] for some special Gibbsian line ensembles, such as the KPZ one (as
we explain below, our results directly imply an alternative proof of this KPZ
result; see 25.1).

Second, (1.1) only requires the limiting trajectory of $\mathcal{L}_{1}(t)$ to
approximate a parabola, as opposed to stipulating it to be exactly
translation-invariant upon a parabolic shift. In agreement with the
terminology from [116, Section 10.4], one might therefore refer to 2.9 as a
_strong characterization_ for the Airy line ensemble. Strong characterizations
for Gibbs measures of random surface models, with power law correlation decay,
appear to be quite rare in the literature (outside of the fairly distant
setting of random lozenge tilings [2]).

Before continuing, let us briefly comment on two potential applications that
our characterization may lead to in the future (for which both of the above-
mentioned improvements would seem to be quite useful). The first concerns
stochastic growth models; many such systems proven to be in the KPZ
universality class are not fully solvable in the sense of being determinantal,
but instead satisfy a Yang–Baxter equation. These include the stochastic six-
vertex model [69, 22] and its degenerations (which encompass the KPZ equation
and ASEP); certain random polymers [101, 113, 38, 13]; $q$-deformations of the
TASEP [112, 21] and PNG model [5]; and various other systems. For all of these
models, it is known that the one-point marginals of their height functions
under wedge initial data converge to the Tracy–Widom GUE distribution;
however, their full convergence to the Airy2 process is still open for most of
them, except for the ASEP, KPZ equation, and O’Connell–Yor polymer [109, 120].
Using the Yang–Baxter equation alone, it is possible to map the height
functions for all of the above models to the arctic boundary of an associated
Gibbsian line ensemble; this was first done for the stochastic six-vertex
model in [20, 32], and later systematized to other models through the
bijectivization framework222The use of this framework (and its special case
called stochasticization [4]) to produce line ensembles from stochastic models
with a Yang–Baxter equation will be elaborated and extended upon in
forthcoming work [3]. of Bufetov, Mucciconi, and Petrov [26, 25]. One-point
convergence results for these models verify the tightness and curvature
assumptions for the top curves of these ensembles, which might enable one to
extend the frameworks developed in [51, 14, 50, 114] to show that they
converge to Brownian Gibbsian line ensembles. Our characterization 2.9 would
then apply, proving their convergence to the Airy line ensemble, and hence of
their top curves (tracking the height function of the associated stochastic
growth model) to the Airy2 process. In Section 25 below, we provide the very
quick implementation of this idea for two examples (where the qualitative
framework has already been set up), namely, the KPZ equation (25.1) and log-
gamma polymer (25.2).

The second potential use of our characterization result is towards proving
convergence of edge statistics for random surfaces to the Airy line ensemble.
At the moment, there seem to be few (if any) natural examples of non-
determinantal random surface models for which this statement has been
proven.333Even one-point convergence statements seem to be rare in this
context, but see the recent work of Ayyer–Chhita–Johansson [11] for such a
result at the edge of the domain-wall ice model. As mentioned previously,
there exists a fairly robust framework [51, 14, 50, 114] for proving
convergence of edge limits of random surfaces to Brownian line ensembles,
assuming certain tightness and curvature constraints for the extreme level
line. An obstruction that remains is thus in verifying these constraints; they
can be reformulated as a weak local law444This terminology was adopted from
works of Erdős–Schlein–Yau [56, 55], showing local semicircle laws for random
matrices. at the facet edge for general random surfaces, meaning that their
limit shape phenomena hold not only on global scales, but also on mesoscopic
ones (of dimensions $n^{1/3}\times n^{2/3}$) near the edge. In the bulk of the
liquid region, such local laws have been proven for random tilings [2] by an
inductive application of the associated variational principle on progressively
smaller scales. It is an enticing question to see if those ideas can be
extended to the facet edge of general random surface models, which together
with our characterization might lead to universality results for the Airy line
ensemble.

We now return to the characterization 2.9 and proceed to describe some of the
ideas behind its proof (see Section 3 below for a more precise exposition).

####  1.2. Proof Overview

In what follows, we denote the parabolic Airy line ensemble by
$\bm{\mathcal{R}}=(\mathcal{R}_{1},\mathcal{R}_{2},\ldots)$. To be consistent
with previous works, it will be its rescaling $2^{-1/2}\cdot\bm{\mathcal{R}}$
that satisfies the Brownian Gibbs property (of variance $1$). So, we will show
that a line ensemble $\bm{\mathcal{L}}$ satisfying (1.1) is equal to
$2^{-1/2}\cdot\bm{\mathcal{R}}$, up to an affine shift. To this end, we will
prove a sequence of results indicating that $\bm{\mathcal{L}}$ is close to
$2^{-1/2}\cdot\bm{\mathcal{R}}$, in an increasingly fine sense. To explain
this further, we first recall from work of Soshnikov [117] (see also that of
Dauvergne–Virág [45]) that with high probability the paths in
$2^{-1/2}\cdot\bm{\mathcal{R}}$ satisfy

(1.2) $\displaystyle
2^{-1/2}\cdot\mathcal{R}_{j}(t)+2^{-1/2}t^{2}=-2^{-7/6}(3\pi)^{2/3}j^{2/3}-\mathcal{O}(j^{-1/3}).$

In particular, given an integer $n\geq 1$, we likely have
$2^{-1/2}\cdot\mathcal{R}_{n}(0)$ is of order $-n^{2/3}$. More generally,
(1.2) implies that $2^{-1/2}\cdot\mathcal{R}_{j}(t)$ is of order $-n^{2/3}$
likely holds, if $t$ is of order $n^{1/3}$ and $j$ is of order $n$. For this
reason, we will compare the top $n$ curves of $\bm{\mathcal{L}}$ to those of
$2^{-1/2}\cdot\bm{\mathcal{R}}$ when the time $t$ and space $x$ parameters are
of order $n^{1/3}$ and $n^{2/3}$, respectively; see Figure 1.2 for a
depiction.

Figure 1.2. Shown above is an order $n^{1/3}\times n^{2/3}$ time-space $(t,x)$
scaling window to examine the curves $\mathcal{L}_{k}$ with $k$ of order $n$.

On a more refined level, (1.2) indicates that, while the “deep” paths (of high
index) in $2^{-1/2}\cdot\bm{\mathcal{R}}$ are of large magnitude, they tightly
concentrate around smooth, deterministic functions, in both the time
(horizontal) and space (vertical) directions. In the time direction, (1.2)
implies that with high probability $2^{-1/2}\cdot\mathcal{R}_{n}(t)$ closely
approximates a parabola of curvature $-2^{-1/2}$, namely,

(1.3)
$\displaystyle\big{|}2^{-1/2}\cdot\mathcal{R}_{n}(t)-\mathfrak{f}_{n}(t)\big{|}=o(1),\qquad\text{for
$\mathfrak{f}_{n}(t)=-2^{-1/2}t^{2}-2^{-7/6}(3\pi)^{2/3}n^{2/3}$}.$

In the space direction, (1.2) implies for any $t$ that
$2^{-1/2}\cdot\big{(}\mathcal{R}_{n+1}(t),\mathcal{R}_{n+2}(t),\ldots,\mathcal{R}_{2n}(t)\big{)}$,
obtained from the paths of $2^{-1/2}\cdot\bm{\mathcal{R}}$ at time $t$ with
indices in $\\{n+1,n+2,\ldots,2n\\}$, likely approximates a smooth,
deterministic profile. More specifically, for any index $k\in[1,n]$, we likely
have

(1.4)
$\displaystyle\big{|}2^{-1/2}\cdot\mathcal{R}_{k+n}(t)-n^{2/3}\cdot\mathfrak{g}_{tn^{-1/3}}(kn^{-1})\big{|}=o(1),\quad\text{for
$\mathfrak{g}_{r}(y)=-2^{-7/6}(3\pi)^{2/3}(y+1)^{2/3}-2^{-1/2}r^{2}$},$

where we observe that the profile $\mathfrak{g}_{r}(y)$ is smooth in
$y\in[0,1]$. To establish 2.9, we will first prove that weaker variants of the
bounds (1.2), (1.3), and (1.4) hold for $\bm{\mathcal{L}}$. In the first, we
allow for a larger error; in the last two, we replace the deterministic
functions $\mathfrak{f}_{n}$ and $\mathfrak{g}_{r}$ with unspecified, random
functions $h_{n}$ and $\gamma_{r}$ (that likely satisfy some similar
properties to $\mathfrak{f}_{n}$ and $\mathfrak{g}_{r}$, respectively).

In particular, we will proceed by proving the following four, increasingly
precise, statements. In what follows, $A>1$ is an arbitrary constant; $n$ is
an integer parameter that we view as tending to infinity;
$t\in[-An^{1/3},An^{1/3}]$ is a time parameter; $i$ and $j$ are indices with
$1\leq i\leq j\leq An$; and all claims below hold with high probability.

  1. (1)

_On-scale estimates_ : The scaling in (1.2) is valid for $\bm{\mathcal{L}}$,
in two senses (3.8).

    1. (a)

_Path locations_ :
$-1500j^{2/3}\leq\mathcal{L}_{j}(t)+2^{-1/2}t^{2}\leq-j^{2/3}/15000$ likely
holds, if $j\geq n/A$.

    2. (b)

_Gap upper bound_ :
$\big{|}\mathcal{L}_{i}(t)-\mathcal{L}_{j}(t)\big{|}\leq\mathcal{O}(j^{2/3}-i^{2/3})+(\log
n)^{25}i^{-1/3}$ likely holds.

  2. (2)

_Global law and regularity_ : For $\bm{\mathcal{L}}$, (1.2) likely holds but
with a larger error $o(n^{2/3})$ (3.10), and (1.4) likely holds but with an
unknown, regular function $\gamma_{r}$ replacing $\mathfrak{g}_{r}$ (3.12).

    1. (a)

_Global law_ :
$\big{|}\mathcal{L}_{j}(t)+2^{1/2}t^{2}+2^{-7/6}(3\pi)^{2}j^{2/3}\big{|}=o(n^{2/3})$
likely holds.

    2. (b)

_Spatial regularity_ : There likely is an almost smooth (whose first $50$
derivatives are uniformly bounded in $n$), random function
$\gamma_{tn^{-1/3}}:[0,1]\rightarrow\mathbb{R}$, such that
$\big{|}\mathcal{L}_{k+n}(t)-n^{2/3}\cdot\gamma_{tn^{-1/3}}(k/n)\big{|}=o(1)$
whenever $1\leq k\leq n$.

  3. (3)

_Curvature approximation_ : There likely is a random function
$h_{n}:[-An^{1/3},An^{1/3}]\rightarrow\mathbb{R}$, so that
$\big{|}h_{n}^{\prime\prime}(s)+2^{-1/2}\big{|}=o(1)$ and
$\big{|}\mathcal{L}_{n}(s)-h_{n}(s)\big{|}=o(1)$ for all $|s|\leq An^{1/3}$
(3.14).

  4. (4)

_Airy statistics_ : The ensemble $\bm{\mathcal{L}}$ has Airy statistics (3.18
and 3.19).

    1. (a)

_Airy gaps_ : The joint law of the gaps
$\big{(}\mathcal{L}_{1}(s)-\mathcal{L}_{2}(s),\mathcal{L}_{2}(s)-\mathcal{L}_{3}(s),\ldots\big{)}$
coincides with those of the Airy point process
$2^{-1/2}\cdot\big{(}\mathcal{R}_{1}(0)-\mathcal{R}_{2}(0),\mathcal{R}_{2}(0)-\mathcal{R}_{3}(0),\ldots\big{)}$.

    2. (b)

_Airy line ensemble_ : Up to an affine shift, $\bm{\mathcal{L}}$ is a
parabolic Airy line ensemble $2^{-1/2}\cdot\bm{\mathcal{R}}$.

To ease the exposition, we will implement the above four tasks out of order.
After providing a more detailed proof discussion and reviewing miscellaneous
preliminary results in this Chapter 1, we will show the on-scale estimates in
Chapter 2. After proving several results about limit shapes for non-
intersecting Brownian bridges in Chapter 4 and couplings for them in Chapter
5, we will establish the global law and regularity for $\bm{\mathcal{L}}$ in
Chapter 6. We will prove the curvature approximation in the second half of
Chapter 3, and that $\bm{\mathcal{L}}$ has Airy statistics in the first half
of Chapter 3.

We next describe the above four statements, and some ideas underlying their
proofs, in greater detail. As we will see, an obstacle we will repeatedly face
is the lack of control on the curves $\mathcal{L}_{j}$ of $\bm{\mathcal{L}}$.
Even up until midway through the last (fourth) statement of the above
overview, our estimates on the $\mathcal{L}_{j}$ will be quite poor, unable to
forbid them from fluctuating more than the parabolic Airy line ensemble
itself. On the other hand, over the past twenty-five years, Dyson Brownian
motion (and equivalent familes of non-intersecting Brownian bridges, including
Brownian watermelons as a special case) has been comprehensively understood,
both from the perspective of exact solvability (starting with the works of
Brézin–Hikami [24] and Johansson [79]) and stochastic analysis (see, for
instance, the reviews of Guionnet [67] and Erdős–Yau [57]).

A substantial portion of our analysis is therefore centered on devising a
series of comparisons between the line ensemble $\bm{\mathcal{L}}$ and Dyson
Brownian motion; this will enable us to transfer results about the latter
(that are sometimes already available in the literature, which we will explain
as they arise) to the former. These two systems initially appear to be quite
different and, indeed, the first forms of our comparisons will be fairly
coarse (though sufficient to prove the on-scale estimates, for example).
However, as we continue to learn more about the line ensemble
$\bm{\mathcal{L}}$, we will use the bounds obtained from previous comparisons
to concoct new and improved ones, eventually reaching the level where we can
compare exact Airy statistics.

Let us outline this in more detail. The main purpose of the below outline is
to serve as a guide for readers examining in greater depth the arguments
presented in the body of this paper; on occasion, they may wish to consult
this outline to recall the overarching ideas and intuition underlying these
arguments, to help them navigate through its lengthier details. As such, this
outline may be skimmed or skipped on an initial reading, especially since it
may get a bit involved at some points.

####  1.3. On-Scale Estimates

Before discussing our proof of the on-scale estimates, we first describe a
coupling, called gap monotonicity, that will play an extensive role in many of
our arguments.

##### Gap Monotonicity

Monotone couplings have long been fundamental in the analysis of random
surfaces. In the context of Brownian Gibbsian line ensembles, the most
commonly used such coupling is called _height monotonicity_ , which indicates
that non-intersecting Brownian bridges are increasing in their boundary data.
More precisely, sample two families of $n$ non-intersecting Brownian bridges
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{n})$ and
$\widetilde{\bm{\mathsf{x}}}=(\widetilde{\mathsf{x}}_{1},\widetilde{\mathsf{x}}_{2},\ldots,\widetilde{\mathsf{x}}_{n})$
starting at $n$-tuples $\bm{u}$ and $\widetilde{\bm{u}}$, respectively; ending
at $n$-tuples $\bm{v}$ and $\bm{\widetilde{v}}$, respectively; and conditioned
to stay above lower boundary curves $f$ and $\widetilde{f}$,
respectively.555One can also constrain them to lie below upper boundaries, but
we will not do so in this introductory exposition. Assume that
$\bm{u}\leq\widetilde{\bm{u}}$, that $\bm{v}\leq\widetilde{\bm{v}}$, and that
$f\leq\widetilde{f}$. Then, it is possible to couple $\bm{\mathsf{x}}$ and
$\widetilde{\bm{\mathsf{x}}}$ such that each
$\mathsf{x}_{j}\leq\widetilde{\mathsf{x}}_{j}$. See the left side of Figure
1.3 for a depiction.

Figure 1.3. Shown to the left is a depiction for height monotonicity. Shown to
the right is a depiction for gap monotonicity

In this paper we further require a different type of monotonicity that
compares not the Brownian bridges themselves but the gaps between them. We
refer to this as _gap monotonicity_ , stated as 5.1 below, which indicates
that the gaps between non-intersecting Brownian bridges are increasing in the
gaps of their starting and ending data, and also in the convexity of their
lower boundary curves. More precisely, assume that each
$|u_{i}-u_{j}|\leq|\widetilde{u}_{i}-\widetilde{u}_{j}|$ and
$|v_{i}-v_{j}|\leq|\widetilde{v}_{i}-\widetilde{v}_{j}|$, and also that
$f^{\prime\prime}\leq\widetilde{f}^{\prime\prime}$ (in the sense of
distributions). Then, it is possible to couple $\bm{\mathsf{x}}$ and
$\widetilde{\bm{\mathsf{x}}}$ such that each
$|\mathsf{x}_{i}-\mathsf{x}_{j}|\leq|\widetilde{\mathsf{x}}_{i}-\widetilde{\mathsf{x}}_{j}|$.
See the right side of Figure 1.3 for a depiction.

Perhaps the simplest proof of height monotonicity (see [34], for example)
proceeds by first discretizing the Brownian bridges into non-intersecting
Bernoulli random walks; coupling the latter under a local Markov chain (such
as the Glauber dynamics) that preserves height orderings; running this chain
until it mixes; and taking any limit point of the dynamics as a height
monotone coupling. Such a proof cannot apply for gap monotonicity, as it is
false in this discrete setup (see 5.6).

To prove gap monotonicity, we instead proceed by first “semi-discretizing” the
Brownian bridges into Gaussian random walks that are continuous in space but
discrete in time. They constitute $T\in\mathbb{Z}_{\geq 1}$ steps, which
allows us to induct on $T$. To this end, we introduce a non-local Markov
chain, which alternates between resampling the first step of all walks
simultaneously and their remaining $T-1$ steps. Using the inductive hypothesis
(replacing $T$ by either $2$ or $T-1$), we show that we can couple these
dynamics so as to preserve gap orderings. By again running this chain until it
mixes, this reduces proving semi-discrete gap monotonicity to verifying its
base case $T=2$, which is done directly, by induction on the number of paths.
See Section 5 below for further details.

##### Path Location Bounds

The first aspect of the on-scale estimates, described in Section 1.2, states
that for $t\in[-An^{1/3},An^{1/3}]$ and $n/A\leq j\leq An$ we likely have

(1.5)
$\displaystyle-1500j^{2/3}\leq\mathcal{L}_{j}(t)+2^{-1/2}t^{2}\leq-\displaystyle\frac{j^{2/3}}{15000}.$

This estimates the deep curves of $\bm{\mathcal{L}}$, only assuming the bound
(1.1) on its top curve. While many of the previously mentioned works on
Gibbsian line ensembles do show some control on these deep curves
$\mathcal{L}_{j}$, their bounds are usually not optimal (a large power, if not
exponential) in their dependence on the index $j$. For certain specific
ensembles relating to last passage percolation models, the true dependence on
$j$ (up to constants, as in (1.5)) was shown by Basu–Ganguly–Hammond–Hegde
[17], by relating such estimates to the geometry of non-intersecting
geodesics.

For general Brownian Gibbsian line ensembles satisfying (1.1), this connection
between $\bm{\mathcal{L}}$ and last passage percolation is lost, and so our
proof instead uses only the Brownian Gibbs property. In particular, we show on
$[-An^{1/3},An^{1/3}]$ that $\mathcal{L}_{j}$ can neither be very high
($\mathcal{L}_{j}(t)+2^{-1/2}t^{2}>-j^{2/3}/15000$), nor very low
($\mathcal{L}_{j}(t)+2^{-1/2}t^{2}<-1500j^{2/3}$). This will be a quick
consequence of combining the following three statements, where in all of them
we assume that $\mathcal{L}_{1}(t)$ is close to the parabola $-2^{-1/2}t^{2}$,
as is likely implied by (1.1). While the statements of, and reasoning behind,
these claims in this exposition will be imprecise, their proper justification
will be obtained by applying height monotonicity to compare $\bm{\mathcal{L}}$
to Brownian watermelons; see Figure 1.4 for depictions and Section 6 below for
further details.

1\. If $\mathcal{L}_{j}$ is very low at a point $t_{0}$, then it is likely low
on a long interval (6.5). Otherwise, there would exist two points
$T_{1}<t_{0}<T_{2}$ not very far from $t_{0}$, such that
$\mathcal{L}_{j}(T_{1})$ and $\mathcal{L}_{j}(T_{2})$ are much higher than
$\mathcal{L}_{j}(t_{0})$. Then resampling the top $j$ curves of
$\bm{\mathcal{L}}$ on $[T_{1},T_{2}]$, one finds that the conditional boundary
data of these $j$ paths is too high to likely allow their bottom curve to drop
to $\mathcal{L}_{j}(t_{0})$ at time $t_{0}$, which is a contradiction. See the
left side of Figure 1.4.

2\. If $\mathcal{L}_{j}$ is very high at a point $t_{0}$, then it is likely
low on a long interval to the right of $t_{0}$ (6.6). Otherwise, there would
exist a point $T>t_{0}$ not very far from $t_{0}$, such that
$\mathcal{L}_{j}(T)$ is not low. Resampling the top $j$ curves of
$\bm{\mathcal{L}}$ on $[t_{0},T]$, one finds that their conditional starting
data at time $t_{0}$ is high enough (and their conditional ending data at time
$T$ is not low enough to counteract them) to cause their top curve
$\mathcal{L}_{1}$ to likely “shoot” far above the parabola $-2^{-1/2}t^{2}$ at
some time $R\in[t_{0},T]$, which contradicts (1.1). See the middle of Figure
1.4.

3\. The curve $\mathcal{L}_{j}$ is likely not low on any long interval
$[T_{1},T_{2}]$ (6.4). Otherwise, resampling the top $j-1$ paths in
$\bm{\mathcal{L}}$ on $[T_{1},T_{2}]$ (and possibly moving their boundary data
up a bit), one finds that the conditional lower boundary $\mathcal{L}_{j}$ is
too low to affect them; as such, it can be removed. Since the interval
$[T_{1},T_{2}]$ is long, in the absence of a lower boundary, the top curve
$\mathcal{L}_{1}$ of these $j-1$ paths will likely stay close to the line
connecting $\mathcal{L}_{1}(T_{1})\approx-2^{-1/2}T_{1}^{2}$ to
$\mathcal{L}_{1}(T_{2})\approx-2^{-1/2}T_{2}^{2}$. Thus, it cannot reach high
enough to meet the parabola $-2^{-1/2}t^{2}$ at, say, the midpoint
$(T_{1}+T_{2})/2$ of $[T_{1},T_{2}]$, which again contradicts (1.1). See the
right side of Figure 1.4.

Figure 1.4. Shown above are the three scenarios discussed in Section 1.3,
where the black curves are of $\bm{\mathcal{L}}$; the red ones are the
watermelons we eventually compare them to; and the orange one is the parabola
that $\mathcal{L}_{1}$ should be close to, by (1.1). On the left,
$\mathcal{L}_{j}$ cannot be too low at time $t_{0}$ (even after pushing some
curves in $\bm{\mathcal{L}}$ down to form the red watermelon). The curve
$\mathcal{L}_{1}$ fails to approximate the orange parabola on the middle
(where it is too high, even after pushing some curves in $\bm{\mathcal{L}}$
down to form the red watermelon) and on the right (where it is too low, even
after pushing some curves in $\bm{\mathcal{L}}$ up to form the red
watermelon).

##### Gap Upper Bound

The second aspect of the on-scale estimates, described in Section 1.2, states
for any $1\leq i\leq j\leq n$ and $t\in[-An^{1/3},An^{1/3}]$ that we likely
have

(1.6)
$\displaystyle\big{|}\mathcal{L}_{i}(t)-\mathcal{L}_{j}(t)\big{|}=\mathcal{O}(j^{2/3}-i^{2/3})+(\log
n)^{25}i^{-1/3}.$

To show this, we imagine that $A\geq 1$ as large (but uniformly bounded); set
$\mathsf{T}=An^{1/3}$; and resample the top $2n$ curves of $\bm{\mathcal{L}}$
on $[-2\mathsf{T},2\mathsf{T}]$, which become non-intersecting Brownian
bridges starting at
$\bm{u}=\big{(}\mathcal{L}_{1}(-2\mathsf{T}),\ldots,\mathcal{L}_{2n}(-2\mathsf{T})\big{)}$;
ending at
$\bm{v}=\big{(}\mathcal{L}_{1}(2\mathsf{T}),\ldots,\mathcal{L}_{2n}(2\mathsf{T})\big{)}$;
and conditioned to stay above $\mathcal{L}_{2n+1}$. By gap monotonicity
(recall Section 1.3), removing the lower boundary $\mathcal{L}_{2n+1}$
increases the gaps between the $\mathcal{L}_{n}$. So, it suffices to prove the
gap upper bound (1.6), but for $2n$ non-intersecting Brownian bridges
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{2n})$
starting at $\bm{u}$ and ending at $\bm{v}$.

At this point, we use a known fact relating the law of the non-intersecting
Brownian bridges $\bm{\mathsf{x}}$ without upper and lower boundary, to that
of certain random matrix spectra; it states the following. For any matrix
$\bm{M}$, let $\operatorname{eig}(\bm{M})$ denote its ordered sequence of
eigenvalues; also set $\bm{U}$ and $\bm{V}$ to be the $2n\times 2n$ diagonal
matrices, with $\operatorname{eig}(\bm{U})=\bm{u}$ and
$\operatorname{eig}(\bm{V})=\bm{v}$. For any $t\in[-2\mathsf{T},2\mathsf{T}]$,
setting $S_{t}=\mathsf{T}-t^{2}/4\mathsf{T}$, the law of $\bm{\mathsf{x}}(t)$
is given by $\operatorname{eig}(\bm{A}_{t}+S_{t}^{1/2}\cdot\bm{G})$, where
$\bm{G}$ is a $2n\times 2n$ GUE random matrix and
$\bm{A}_{t}=(1/2-t/2\mathsf{T})\cdot\bm{U}+(1/2+t/2\mathsf{T})\cdot\bm{W}\bm{V}\bm{W}^{*}$,
for $\bm{W}$ a certain (not Haar distributed) unitary random matrix (see 4.28
below for the precise statement). Hence, the

(1.7) $\displaystyle\text{law of $\bm{\mathsf{x}}(t)$ is given by Dyson
Brownian motion run for time $S_{t}$, with initial data
$\operatorname{eig}(\bm{A}_{t})$}.$

While Dyson Brownian motion is now well understood, effectively using (1.7) is
typically complicated by the involved law of $\bm{A}_{t}$. However, in our
setting, we will only require a bound on the norm of $\bm{A}_{t}$ (after
subtracting a multiple of the identity from it), namely, that we likely have
$\|\bm{A}_{t}-2^{3/2}\mathsf{T}^{2}\cdot\operatorname{Id}\|\leq 1500n^{2/3}$;
this quickly follows from the same bound on $\bm{U}$ and $\bm{V}$, which hold
by the path location estimate (1.5). Thus, by (1.7), the law of
$\bm{\mathsf{x}}(t)$ is given by Dyson Brownian motion, run for time $S_{t}$,
on initial data supported on an interval of width $3000n^{2/3}$.

For $t\in[-An^{1/3},An^{1/3}]=[-\mathsf{T},\mathsf{T}]$, we have
$S_{t}=\mathsf{T}-t^{2}/4\mathsf{T}\geq 3\mathsf{T}/4>An^{1/3}/2$. So, for $A$
large this amounts to running Dyson Brownian motion for a long time, on
initial data supported on a bounded interval.666Under our normalization of
Dyson Brownian motion, $n^{1/3}$ and $n^{2/3}$ are the natural scalings of
time and space, respectively. So, we only take the prefactors $A/2$ and $3000$
into account when using the words, “long” and “bounded.” It is known in this
context that the first $n$ particles equilibrate to have gaps likely
satisfying (1.6) (for example, this sort of statement was shown by
Lee–Schnelli [92]; the slightly improved formulation we use appeared in [6]),
implying the gap upper bound. See Section 7 below (which also includes some
Hölder regularity bounds and improvements of the path location estimates,
which we do not discuss here but will be useful later in the paper) for
further details.

####  1.4. Global Law and Regularity

The proofs of the global law and regularity are based on the notion that non-
intersecting Brownian bridges without lower and upper boundaries are simpler
to analyze than those with them; the relation (1.7) to Dyson Brownian motion
already provides one manifestation of this phenomenon. To realize this idea,
we will restrict the ensemble $\bm{\mathcal{L}}$ to a tall rectangle, giving
rise to a family of non-intersecting Brownian bridges with a lower (and no
upper) boundary; we will then implement two tasks. The first is to introduce a
coupling that compares a family of non-intersecting Brownian bridges on a tall
rectangle with lower boundary, to one with the same starting and ending data
but without a boundary; we refer to it as the boundary removal coupling. The
second is to prove versions of the global law and spatial regularity for non-
intersecting Brownian bridges, without boundary, on a tall rectangle. For the
global law, the latter will require regularity estimates at the edge of
certain limit shapes; we explain this first.

From this point (particularly in this Section 1.4 and the next Section 1.5),
this proof outline will begin becoming more analytically involved.

##### Limit Shapes Near the Edge

It has been known since works of Guionnet and Zeitouni [68, 66] (proving
earlier predictions of Matytsin [94]) that non-intersecting Brownian bridges,
without upper and lower boundaries, exhibit a limit shape phenomenon777By
(1.7), this amounted to a result on Dyson Brownian motion, namely a large
deviations principle for it. in the following sense (see 10.1 below for a more
precise statement, under a slightly different normalization). Fix real numbers
$a<b$ (which will act as times) and $R>0$ (which will parameterize the number
of Brownian bridges). For each integer $n\geq 1$, let
$\bm{u}^{n}=(u_{1},u_{2},\ldots,u_{Rn})$ and
$\bm{v}^{n}=(v_{1},v_{2},\ldots,v_{Rn})$ denote $(Rn)$-tuples, such that
$n^{-2/3}\cdot\bm{u}$ and $n^{-2/3}\cdot\bm{v}$ both converge to some given
profiles. Then letting
$\bm{\mathsf{x}}^{n}=(\mathsf{x}_{1}^{n},\mathsf{x}_{2}^{n},\ldots,\mathsf{x}_{Rn}^{n})$
denote $Rn$ non-intersecting Brownian bridges on $[an^{1/3},bn^{1/3}]$,
starting at $\bm{u}$ and ending at $\bm{v}$, their rescaled trajectories888In
fullest generality, it is technically only the associated height function that
converges in this way, but in this introductory exposition we ignore that
subtlety (which becomes irrelevant in the presence of the gap upper bound).
$n^{-2/3}\cdot\mathsf{x}_{j}(tn^{1/3})$ converge to a limit shape
$G(t,jn^{-1})$, for each $(t,j)\in[a,b]\times[1,Rn]$. Some properties of this
function $G:[a,b]\times[0,R]\rightarrow\mathbb{R}$ are known (see Section 10
below for further details), for example that it satisfies a partial
differential equation on the region where it is smooth, given by

(1.8)
$\displaystyle\partial_{y}^{2}G+(\partial_{y}G)^{-4}\cdot\partial_{t}^{2}G=0.$

The assumption above already underscores the relevance of the on-scale
estimates from Section 1.2. Setting
$\bm{u}=\big{(}\mathcal{L}_{1}(an^{1/3}),\ldots,\mathcal{L}_{Rn}(an^{1/3})\big{)}$
and
$\bm{v}=\big{(}\mathcal{L}_{1}(bn^{1/3}),\ldots,\mathcal{L}_{Rn}(bn^{1/3})\big{)}$,
one can only hope for $n^{-2/3}\cdot\bm{u}$ and $n^{-2/3}\cdot\bm{v}$ to have
(subsequential) limits if some form of (1.5) holds (perhaps with different
constants). The conclusion above here is also reminiscent of the global law
from Section 1.2; both provide deterministic approximations for the Brownian
paths, up to error $o(n^{2/3})$. However, the deterministic approximation
there had an exact formula, but here it is given by the less transparent
function $G$.

While the full limit shape $G$ is usually quite inexplicit indeed, we will
show under certain conditions that it admits a “universal behavior” near the
edge $y=0$ (corresponding to the top curves in $\bm{\mathsf{x}}$).
Specifically, for fixed $\mathfrak{t}\in(a,b)$, there exist constants
$\mathfrak{a},\mathfrak{b}\in\mathbb{R}$ and $\mathfrak{c}>0$ such that

(1.9) $\displaystyle
G(t,y)\approx\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2}-2^{-4/3}(3\pi)^{2/3}\mathfrak{c}^{-1/3}y^{2/3},\qquad\text{for
$(t,y)\approx(\mathfrak{t},0)$}.$

It will be central for the approximation error in (1.9) to remain uniformly
small as the parameter $R$ grows, which we will ultimately take to be large
(to later compare $\bm{\mathcal{L}}$ to a system of Brownian bridges without
boundaries); see 14.1 below (where the $R$ here is $L^{3/2}$ there).

The proof of (1.9) is based on a purely deterministic analysis of the limit
shape $G$ and the associated partial differential equation (1.8). Non-
uniformly elliptic equations similar to (1.8) (though different in that they
were constrained to be Lipschitz, as they arose from limit shapes of dimer
models) were analyzed from a real analytic perspective by De Silva–Savin [47]
and from a complex analytic one by Kenyon–Okounkov [89] and
Astala–Duse–Prause–Zhong [86]. In that setting, the last work [86] proved a
variant of (1.9), though they did not investigate the uniformity of that
approximation in the size of the underlying domain.

In our context, this uniformity is in fact false in general. To witness it, we
must impose hypotheses on the boundary data for $G$ (13.7 and 13.8),
stipulating the existence of a constant $C>1$ so that for each boundary point
$s\in\\{a,b\\}$ we have

(1.10) $\displaystyle-C-Cy^{2/3}\leq G(s,y)\leq
C-C^{-1}y^{2/3},\quad\text{and}\quad\big{|}G(s,y)-G(s,y^{\prime})\big{|}\leq
C\big{|}y^{2/3}-(y^{\prime})^{2/3}\big{|},$

for each $y,y^{\prime}\in[0,R]$. Observe that two bounds in (1.10) constitute
continuum counterparts of the two on-scale estimates (path locations and gap
upper bound) from Section 1.2.

The proof of (1.9) first requires an _a priori_ estimate on how
$\partial_{y}G(t,y)$ diverges for small $y$, namely,
$\partial_{y}G(t,y)\sim-y^{-1/3}$, where the implicit constants are uniform in
$R$ (13.12). To verify the upper bound on $|\partial_{y}G|$, we show a
continuous variant (10.15) of gap monotonicity, indicating that the
$y$-derivatives of limit shapes are increasing in those of their boundary
data. Since (1.10) upper bounds $|\partial_{y}G|$ on the boundary, we can use
this to upper bound the $y$-derivative of $G$ by that of an explicit limit
shape, which can directly be seen to have the $y^{-1/3}$ divergence.

To lower bound $|\partial_{y}G|$, we instead use the following property about
limit shapes [66]. Let $\varrho=-(\partial_{y}G)^{-1}$, fix $t\in(a,b)$, and
set $\tau=(b-t)(t-a)/(b-a)$. Then $\varrho(t,\cdot)$ is the density for a
measure $\nu_{\tau}$, given by the free convolution between some measure $\nu$
of total mass $R$ and the semicircle distribution of size $\tau$. This is
specific to limit shapes for Brownian bridges without upper and lower
boundaries, and it can be viewed as a continuum counterpart of (1.7).
Similarly to (1.7), effectively using this fact is complicated by the fact
that little is in general known about $\nu$.

So, we develop a general estimate for such measures when $\tau\gtrsim 1$
stating that, if the first bound in (1.10) holds at $s=t$, then
$\varrho\lesssim 1$ holds uniformly in $R=\nu(\mathbb{R})$ for $y\leq 1$
(13.3). While the former bound in (1.10) was only stipulated to hold at
$s\in\\{a,b\\}$, it can be shown (by a continuous variant of height
monotonicity, 10.14) to extend to $s\in[a,b]$. This verifies the assumption in
the above free convolution result, yielding for $y\leq 1$ that
$\varrho\lesssim 1$, and so $|\partial_{y}G|\gtrsim 1$. Improving this bound
to $|\partial_{y}G|\gtrsim y^{-1/3}$ requires further effort (involving
elliptic regularity and another application of the continuum gap
monotonicity). See Section 13 below for further details.

Given the above, to establish (1.9), we next use the fact [94] that (1.8) can
be equivalently written as a complex Burgers equation for the complex slope
$f=\partial_{t}G-\mathrm{i}\cdot(\partial_{y}G)^{-1}$, providing $f$ a
holomorphic structure; such ideas were also fruitful in prior works [89, 86]
analyzing dimer limit shapes. In particular, defining the complex coordinate
$z(t,y)=y-t\cdot f(t,y)$, this indicates that $f=F(z)$, for some holomorphic
function $F$. We show that the previously mentioned bounds for $\partial_{y}G$
imply uniform derivative estimates for $F$ (14.5), enabling $F$ to be Taylor
expanded. Translating this expansion for $F$ into one for $G$ eventually
yields the approximation (1.9). See Section 14 below for further details.

##### Boundary Removal Coupling

The boundary removal coupling can be described as follows (see 16.4 below for
the precise statement, where the $R$ here is $L^{3/2}$ there). Fix a bounded
real number $A\geq 1$; let
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ denote a Brownian
Gibbsian line ensemble likely satisfying path location bounds of the type
(1.5); and let $R>1$ be a real number and $n\geq 1$ be an integer (that can
now be arbitrarily large with respect to each other). Sample $Rn$ non-
intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{Rn})$ on
$[-An^{1/3},An^{1/3}]$, starting at
$\bm{u}=\big{(}\mathsf{x}_{1}(-An^{1/3}),\ldots,\mathsf{x}_{Rn}(-An^{1/3})\big{)}$
and ending at
$\bm{v}=\big{(}\mathsf{x}_{1}(An^{1/3}),\ldots,\mathsf{x}_{Rn}(An^{1/3})\big{)}$.
As $\bm{\mathsf{y}}$ has no lower boundary, height monotonicity (recall
Section 1.3) yields a coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$
such that $\mathsf{x}_{j}\geq\mathsf{y}_{j}$ almost surely, for each
$j\in[1,Rn]$; see the right side of Figure 1.5 for a depiciton. We will show
that there exists a coupling satisfying a bound in the reverse direction for
small indices $j$, that is, for $c=2^{-6000}$ we have with high probability
that

(1.11)
$\displaystyle\mathsf{y}_{j}\geq\mathsf{x}_{j}-R^{-c}n^{2/3},\qquad\text{for
each $j\in[1,R^{c}n]$.}$

See the left side of Figure 1.5 for a depiction. Together, these couplings
suggest that the top $R^{c}n$ curves in $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}$ should be close, for large $R$.

Figure 1.5. Depicted above is the boundary removal coupling.

To exhibit the boundary removal coupling, we first reduce it to a “preliminary
coupling” that introduces a lower boundary
$f:[-An^{1/3},An^{1/3}]\rightarrow\mathbb{R}$ for $\bm{\mathsf{y}}$. It
essentially states the following (see 16.9 below for the precise formulation).
Assume that with high probability the path location estimates of the type
(1.5) hold for $\bm{\mathsf{x}}$ and moreover that, (i) $\mathsf{x}_{Rn+1}$ is
not too far above $f$, namely, $f\geq\mathsf{x}_{Rn+1}-(R^{\alpha}n)^{2/3}$
for some $\alpha<1$ and, (ii) its paths $\mathsf{x}_{j}$ are regular, namely,
they satisfy a Hölder bound that is governed by a parameter $\beta\in(0,1)$ in
a specific way (see Definition 16.7 below), where smaller $\beta$ implies
improved regularity. Then, the preliminary coupling between $\bm{\mathsf{x}}$
and $\bm{\mathsf{y}}$ ensures for $c_{0}=2^{-5500}$ that
$\mathsf{y}_{j}\geq\mathsf{x}_{j}-R^{c_{0}(2\beta-7/8)}n^{2/3}$ likely
holds,999Here, the constant $2\beta-7/8$ in the exponent is not optimal. It
can at least be improved to $2\beta-1^{-}$, but we do not know what the
optimal constant should be. for each $j\in[1,R^{c_{0}}n]$.

For the original line ensemble $\bm{\mathsf{x}}$ in the boundary removal
coupling, we will show that (ii) holds at $\beta=3/8$ (16.13), so
$c_{0}(2\beta-7/8)=-c_{0}/8<-c$, yielding the negative exponent in (1.11).
This $\beta=3/8$ result is established in Section 18 below, and its proof
amounts to an inductive series of comparisons between $\bm{\mathsf{x}}$ and
Dyson Brownian motion, the latter of whose regularity properties are well
understood (some of which are collected in Section 15 below). Since
$\bm{\mathsf{y}}$ has no lower boundary, (i) cannot be literally be true as
written, but we may view its bottom path $\mathsf{y}_{Rn}$ as a lower boundary
for its remaining $Rn-1$ ones
$(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{Rn-1})$. A weak estimate on
how much $\mathsf{x}_{Rn}$ and $\mathsf{y}_{Rn}$ can oscillate on the interval
$[-An^{1/3},An^{1/3}]$ (recall $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ share
the same endpoints) will suffice to show that
$|\mathsf{x}_{Rn}-\mathsf{y}_{Rn}|\leq(R^{\alpha}n)^{2/3}$ holds with high
probability for some $\alpha<1$. This enables us to deduce the boundary
removal coupling as a consequence of the preliminary one, applied to
$(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{Rn-1})$ with lower boundary
$f=\mathsf{y}_{Rn}$. See Section 16 below for further details.

It remains to prove that the preliminary coupling exists, which can
heuristically be explained through a diffusive scaling. Fix some parameter
$\vartheta\sim R^{2(\alpha-1)/3}$ and define the non-intersecting Brownian
bridges
$\bm{\mathsf{z}}=(\mathsf{z}_{1},\mathsf{z}_{2},\ldots,\mathsf{z}_{Rn})$ with
lower boundary $\widetilde{f}$, by first diffusively “shrinking”
$\bm{\mathsf{y}}$ (with lower boundary $f$) by factors of $1-\vartheta$ in
space and $(1-\vartheta)^{2}$ in time, and then by translating them up
slightly. See Figure 1.6 for a depiction. This has two effects. First, it can
be shown that the original lower boundary $f\sim-(Rn)^{2/3}$ is quite low, so
shrinking it to $\widetilde{f}$ lifts it up considerably, by
$\vartheta\cdot|f|\sim(R^{\alpha}n)^{2/3}$. Due to (i), this (upon the proper
choice of constants) pulls this lower boundary above $\mathsf{x}_{Rn+1}$.
Second, it changes the time domain of the bridges slightly, from
$[-An^{1/3},An^{1/3}]$ to
$\big{[}-An^{1/3}(1-\vartheta)^{2},An^{1/3}(1-\vartheta)^{2}\big{]}$. By the
Hölder bound (ii), the paths in $\bm{\mathsf{x}}$ cannot increase much when
passing from the former interval to the latter one. Hence, the starting and
ending data for $\bm{\mathsf{z}}$ on
$\big{[}-An^{1/3}(1-\vartheta)^{2},An^{1/3}(1-\vartheta)^{2}\big{]}$ likely
continues to dominate that of $\bm{\mathsf{x}}$.

Figure 1.6. Shown above is a depiction for the proof of the preliminary
coupling described in Section 1.4.

Height monotonicity then provides a coupling between $\bm{\mathsf{x}}$ and
$\bm{\mathsf{z}}$ such that $\mathsf{z}_{j}\geq\mathsf{x}_{j}$ for each
$j\in[1,Rn]$. Since the top curves of $\bm{\mathsf{y}}$ are only barely
perturbed under the shrinking to $\bm{\mathsf{z}}$, this coupling lower bounds
the top curves of $\bm{\mathsf{y}}$ by those of $\bm{\mathsf{x}}$. More
specifically, we can deduce for some explicit constants $d=d(\alpha,\beta)>0$
and $\Delta=\Delta(\alpha,\beta)>0$ that
$\mathsf{y}_{j}\geq\mathsf{x}_{j}-(R^{d(\alpha-\Delta)}n)^{2/3}$ for each
$j\in[1,R^{d}n+1]$ (17.2). Replacing $R$ by $R^{d}$, this effectively reduces
the original exponent $\alpha$ appearing in the preliminary coupling to
$\alpha-\Delta$. While this procedure might not immediately yield an exponent
of $2\beta-7/8$, it can be applied recursively. By repeating it many times, we
can reduce $\alpha$ to just above the value $\alpha_{0}$ where
$\Delta(\alpha_{0},\beta)=0$. A calculation reveals that
$\alpha_{0}<2\beta-7/8$ (17.3), which yields the exponent stated in the
preliminary coupling. See Section 17 below for further details.

##### Global Law

The global law from Section 1.2 states that, for any fixed $\delta>0$,

(1.12)
$\displaystyle\big{|}\mathcal{L}_{n}(t)+2^{-1/2}t^{2}+2^{-7/6}(3\pi)^{2/3}n^{2/3}\big{|}<\delta
n^{2/3},$

likely holds. To establish it, we let $\theta\in(0,1)$ and $R>1$ be small and
very large (relative to $\delta$) real numbers, respectively, both of which
are fixed in $n$ (we also assume $R$ is much larger than $\theta^{-1}$).
Letting $N=n/\theta^{3}$, we then apply the boundary removal coupling (recall
Section 1.4) to the top $RN$ paths of $\bm{\mathcal{L}}$. Sampling $RN$ non-
intersecting Brownian bridges without boundaries
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{RN})$ on
$[-N^{1/3},N^{1/3}]$, starting at
$\bm{u}=\big{(}\mathcal{L}_{1}(-N^{1/3}),\ldots,\mathcal{L}_{RN}(-N^{1/3})\big{)}$
and ending at
$\bm{v}=\big{(}\mathcal{L}_{1}(N^{1/3}),\ldots,\mathcal{L}_{RN}(N^{1/3})\big{)}$,
this enables us to approximate the upper $N\geq n$ paths $\bm{\mathcal{L}}$ by
those of $\bm{\mathsf{y}}$, up to an error of $R^{-c}N^{2/3}\ll\delta
n^{2/3}$. See the left side of Figure 1.7 for a depiction.

Figure 1.7. Shown to the left is a depiction of how the boundary removal
coupling is used to prove the global law; shown to the right is a depiction of
how it is used to prove the spatial regularity.

We next apply limit shape results from [68, 66] to $\bm{\mathsf{y}}$ (recall
Section 1.4). Although they were originally only stated for infinite sequences
of Brownian bridge ensembles with starting and ending data that “converge,” a
compactness argument can be used to show a finite variant of it (20.3). This
provides a limit shape $G:[-1,1]\times[0,1]\rightarrow\mathbb{R}$ such that we
likely have $\big{|}\mathsf{y}_{j}(tN^{1/3})-N^{2/3}\cdot
G(t,j/N)\big{|}\ll\delta n^{1/3}$. Our edge behavior result described in
Section 1.4, whose hypothesis (1.10) can be verified by the two on-scale
estimates from Section 1.2, then applies to this limit shape $G$ and yields
(at this point, unknown) constants $(\mathfrak{a},\mathfrak{b},\mathfrak{c})$
so that

$\displaystyle
G(s,y)\approx\mathfrak{a}+\mathfrak{b}s-\mathfrak{c}s^{2}-2^{-4/3}(3\pi)^{2/3}\mathfrak{c}^{-1/3}y^{2/3},\qquad\text{for
small $(s,y)\in[-\theta,\theta]\times[0,\theta]$}.$

Combining this at $(s,y)=(\theta t,j/N)$ with the previous statements, we
deduce

$\displaystyle\big{|}\mathcal{L}_{j}(tn^{1/3})-n^{2/3}\cdot(\theta^{-2}\mathfrak{a}+\theta^{-1}\mathfrak{b}t-\mathfrak{c}t^{2})+2^{-4/3}(3\pi)^{2/3}\mathfrak{c}^{-1/3}j^{2/3}\big{|}\ll\delta
n^{2/3},$

likely holds for $j\leq n\leq\theta N$. Matching this against the behavior
(1.1) imposed on the top curve $j=1$, we obtain
$(\mathfrak{a},\mathfrak{b},\mathfrak{c})=(0,0,2^{-1/2})$, giving (1.12). See
Section 20 below for further details.

##### Spatial Regularity

For any $t\in[-A,A]$, the spatial regularity from Section 1.2 exhibits a
random, almost smooth (say, with bounded first $50$ derivatives) function
$\gamma_{t}:[0,1]\rightarrow\mathbb{R}$ such that

(1.13)
$\displaystyle\big{|}\mathcal{L}_{j+n}(tn^{1/3})-n^{2/3}\cdot\gamma_{t}(jn^{-1})\big{|}=o(1),\qquad\text{likely
holds for each $1\leq j\leq n$}.$

This provides a much stronger approximation than the global law (1.12), at the
expense of making the approximating function $\gamma_{t}$ less explicit. Its
proof again makes use of the boundary removal coupling (recall Section 1.4),
but now takes $R\gg n^{-2/c}$ (at $c=2^{-6000}$) to grow much faster than $n$.
Sampling $Rn$ non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{Rn})$ on
$[-2An^{1/3},2An^{1/3}]$, starting at
$\bm{u}=\big{(}\mathcal{L}_{1}(-2An^{1/3}),\ldots,\mathcal{L}_{Rn}(-2An^{1/3})\big{)}$
and ending at
$\bm{v}=\big{(}\mathcal{L}_{1}(2An^{1/3}),\ldots,\mathcal{L}_{Rn}(2An^{1/3})\big{)}$,
this approximates the upper $R^{c}n\gg 2n$ paths of $\bm{\mathcal{L}}$ by
those of $\bm{\mathsf{y}}$, up to an error of $R^{-c}n^{2/3}=o(1)$. See the
right side of Figure 1.7 for a depiction. It thus suffices prove spatial
regularity for $\bm{\mathsf{y}}$.

The benefit in this is that, since $\bm{\mathsf{y}}$ does not have an upper or
lower boundary, it admits the interpretation (1.7) in terms of Dyson Brownian
motion. Rigidity results of Huang–Landon [76] apply to the latter and imply
that the $\mathsf{y}_{j}(t)$ closely concentrate, up to error $o(1)$, around
the quantiles of a measure $\nu_{\tau}$, given by the free convolution between
some measure $\nu$ of total mass $R$ and the semicircle distribution of size
$\tau=A-t^{2}/4A\gtrsim 1$. The spatial regularity of $\bm{\mathsf{y}}$ then
amounts to ensuring that the density for this measure $\nu_{\tau}$ is almost
smooth, but this is once again complicated by the fact that little is known
about $\nu$. So, we develop a general result about such measures $\nu_{\tau}$,
closely related to the one described in Section 1.4. It states that, under
certain conditions (which can in our context can be later verified by the two
on-scale estimates described in Section 1.2), when $\tau\gtrsim 1$ the
derivatives of the density for $\nu_{\tau}$ are uniformly bounded in
$R=\nu(\mathbb{R})$ (13.4). This confirms the spatial regularity for
$\bm{\mathsf{y}}$ and thus for $\bm{\mathcal{L}}$. See Section 19 below for
further details.

####  1.5. Curvature Approximation

The curvature approximation from Section 1.2 exhibits a random, twice-
differentiable function $h_{n}:[-An^{1/3},An^{1/3}]\rightarrow\mathbb{R}$ so
that we likely have

(1.14)
$\displaystyle\big{|}h_{n}^{\prime\prime}(s)+2^{-1/2}\big{|}=o(1),\quad\text{and}\quad\big{|}h_{n}(s)-\mathcal{L}_{n}(s)\big{|}=o(1),\qquad\text{for
all $s\in[-An^{1/3},An^{1/3}]$}.$

As for spatial regularity (recall Section 1.4), it provides a stronger
approximation than the global law (1.12), though with a less explicit
approximating function $h_{n}$. To establish it, we make use of a
concentration bound for non-intersecting Brownian bridges with smooth boundary
data, proven in [8, Sections 4-7] (based on carefully “patching together”
concentration bounds for Dyson Brownian motion, inspired by ideas of
Laslier–Toninelli [91]). That bound can be described as follows (see 10.27
below for a more precise statement, under a slightly different normalization).

Let $a<b$ be real numbers; let $k\geq 1$ be a large integer (which we view as
tending to $\infty$); and let
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{k})$ denote
non-intersecting Brownian bridges on $[ak^{1/3},bk^{1/3}]$, starting at
$\bm{u}=(u_{1},u_{2},\ldots,u_{k})$; ending at
$\bm{v}=(v_{1},v_{2},\ldots,v_{k})$; and conditioned to lie above and below
functions $f:[ak^{1/3},bk^{1/3}]\rightarrow\mathbb{R}$ and
$g:[ak^{1/3},bk^{1/3}]\rightarrow\mathbb{R}$, respectively. Assume that there
is an almost smooth (with bounded first $50$ derivatives) solution
$G:[a,b]\times[0,1]\rightarrow\mathbb{R}$ to the limit shape partial
differential equation (1.8), which are close to $(\bm{u};\bm{v};f;g)$ along
the boundary, namely, for each $j\in[1,k]$ and $t\in[a,b]$ we have

(1.15) $\displaystyle\begin{aligned} &\big{|}k^{2/3}\cdot
G(a,jk^{-1})-u_{j}\big{|}=o(1);\qquad\quad\big{|}k^{2/3}\cdot
G(b,jk^{-1})-v_{j}\big{|}=o(1);\\\ &k^{2/3}\cdot
G(t,0)=g(tk^{1/3});\qquad\qquad\qquad k^{2/3}\cdot
G(t,1)=f(tk^{1/3}).\end{aligned}$

Then, $\big{|}\mathsf{x}_{j}(tk^{1/3})-k^{2/3}\cdot G(t,jk^{-1})\big{|}=o(1)$
holds for all $(t,j)\in[a,b]\times[1,k]$, with high probability.

The condition (1.15) can be viewed as a constraint on the boundary data
$(\bm{u};\bm{v};f;g)$ for $\bm{\mathsf{x}}$, as it implies that they must
approximate a smooth function $G$. In particular, the first two bounds in
(1.15) underscores the relevance of spatial regularity for $\bm{\mathcal{L}}$.
Fixing $k=2n/3$ and letting
$\bm{u}=\big{(}\mathcal{L}_{k+1}(-Ak^{1/3}),\ldots,\mathcal{L}_{2k}(-Ak^{1/3})\big{)}$
and
$\bm{v}=\big{(}\mathcal{L}_{k+1}(Ak^{1/3}),\ldots,\mathcal{L}_{2k}(Ak^{1/3})\big{)}$,
they can only hold if (1.13) does (with the $n$ equal to $k$ here) for an
almost smooth function $\gamma_{t}(\cdot)=G(t,\cdot)$, at $t\in\\{-A,A\\}$.

While this spatial regularity ensures that the starting and ending data
$(\bm{u};\bm{v})$ approximate almost smooth profiles, it makes no such
guarantee for the upper and lower boundaries $(f;g)$. In our case, the latter
are $(\mathcal{L}_{k};\mathcal{L}_{2k+1})$, and we do not know how to directly
show that they are close to smooth functions. To circumvent this issue, we let
$\mathfrak{w}\in(0,1)$ be a small parameter and subdivide
$[-An^{1/3},An^{1/3}]$ into subintervals $\\{I_{j}\\}$ of length
$2\mathfrak{w}n^{1/3}$. For each $j\in[1,A/\mathfrak{w}]$, we will produce a
“local” approximating function $h_{n;j}$ likely satisfying (1.14) on $I_{j}$
(11.2), and then we will “glue” these local $h_{n;j}$ together (11.3) to form
a global one satisfying (1.14) on $[-An^{1/3},An^{1/3}]$. See the left side of
Figure 1.8 for a depiction. This reduces us to proving a version of (1.14),
but only on short intervals of length $2\mathfrak{w}n^{1/3}$. The value in
this is that, on such thin domains, one does not expect the upper and lower
boundaries $f=\mathcal{L}_{k}$ and $g=\mathcal{L}_{2k+1}$ to substantially
affect most of the middle curves.

Figure 1.8. Shown to the left is the gluing used to produce a global $h_{n}$
satisfying (1.14). Shown to the right is the sandwiching of $\bm{\mathcal{L}}$
between $\bm{\mathcal{L}}^{+}$ and $\bm{\mathcal{L}}^{-}$.

To make this precise, we introduce two families $\bm{\mathcal{L}}^{-}$ and
$\bm{\mathcal{L}}^{+}$ of non-intersecting Brownian bridges and sandwich
$\bm{\mathcal{L}}$ between them. Their starting and ending data will nearly
coincide with those of $\bm{\mathcal{L}}$, which are almost smooth by spatial
regularity (1.13). Their (upper and lower) boundaries will also be smooth, and
will lie slightly above and below those of $\bm{\mathcal{L}}$, respectively.
See the right side of Figure 1.8 for a depiction. This regularity will enable
us to approximate (as in (1.15)) the boundary data for $\bm{\mathcal{L}}^{-}$
and $\bm{\mathcal{L}}^{+}$ by almost smooth limit shapes $G^{-}$ and $G^{+}$
satisfying (1.8), respectively. By the global law (1.12), the boundary data
for $G^{-}$ and $G^{+}$ approximate the function
$\mathfrak{G}(t,y)=-2^{-1/2}t^{2}-2^{-7/6}(3\pi)^{2/3}y^{2/3}$. Using
properties of solutions to the equation (1.8), we will further show (11.4)
that these limit shapes essentially satisfy, (i) they are extremely close to
each other in the middle, so
$\big{|}G^{+}(t,1/2)-G^{-}(t,1/2)\big{|}=o(k^{-2/3})$ and, (ii) they
approximate $\mathfrak{G}$ also in their derivatives, so
$\partial_{t}^{2}G^{+}=\partial_{t}^{2}\mathfrak{G}+o(1)=o(1)-2^{-1/2}$ and
similarly $\partial_{t}^{2}G^{-}=o(1)-2^{-1/2}$.

Thus the above concentration bound applies to $\bm{\mathcal{L}}^{-}$ and
$\bm{\mathcal{L}}^{+}$, which with the sandwiching of $\bm{\mathcal{L}}$
between them gives $k^{2/3}\cdot
G^{-}(t,1/2)-o(1)\leq\mathcal{L}_{n}^{-}(tk^{1/3})\leq\mathcal{L}_{n}(tk^{1/3})\leq\mathcal{L}_{n}^{+}(tk^{1/3})=k^{2/3}\cdot
G^{+}(t,1/2)+o(1)$ with high probability. By (i), the left and right sides of
this inequality are within $o(1)$ of each other, and so
$\mathcal{L}_{n}(tk^{1/3})=k^{2/3}\cdot G^{+}(t,1/2)+o(1)$. Then taking
$h_{n;j}(t)=k^{2/3}\cdot G^{+}(tk^{-1/3},1/2)$ yields the second statement in
(1.14). By (ii), we also have
$h_{n;j}^{\prime\prime}(t)=\partial_{t}^{2}G^{+}(tk^{-1/3},1/2)=o(1)-2^{-1/2}$,
confirming the first statement in (1.14). See Section 11 and Section 12 below
for further details, where the proper implementation of the above framework
involves an induction on scales argument.

####  1.6. Airy Statistics

Although we now have the curvature approximation (1.14), we are not yet able
to directly compare $\bm{\mathcal{L}}$ to the parabolic Airy line ensemble
$2^{-1/2}\cdot\bm{\mathcal{R}}$. Indeed, the bound there
$|h_{n}^{\prime\prime}+2^{-1/2}|=o(1)$ on the approximating function $h_{n}$
still in principle allows it to have large oscillations, of sizes up to
$o(n^{2/3})$ on the time interval $[-n^{1/3},n^{1/3}]$; these can already
dominate the fluctuations of $2^{-1/2}\cdot\bm{\mathcal{R}}$. We instead first
pin down a more robust family of statistics for $\bm{\mathcal{L}}$ (as opposed
to its entire law), given by its gaps
$\big{(}\mathcal{L}_{1}(t)-\mathcal{L}_{2}(t),\mathcal{L}_{2}(t)-\mathcal{L}_{3}(t),\ldots\big{)}$
at a given time $t$.

##### Airy Gaps

The first aspect of Airy statistics, as described in Section 1.2, states that
for any fixed $t\in\mathbb{R}$ the law of
$\big{(}\mathcal{L}_{1}(t)-\mathcal{L}_{2}(t),\mathcal{L}_{2}(t)-\mathcal{L}_{3}(t),\ldots\big{)}$
coincides with that of
$2^{-1/2}\cdot\big{(}\mathcal{R}_{1}(0)-\mathcal{R}_{2}(0),\mathcal{R}_{2}(0)-\mathcal{R}_{3}(0),\ldots\big{)}$.
To establish this, we show that the former stochastically is lower bounded by
the latter (8.2), and also is stochastically upper bounded by it (8.3). The
proofs of both use gap monotonicity (recall Section 1.3) in different ways.

To prove the lower bound, we fix a large integer $n\gg 1$ and make use of the
curvature approximation (1.14), recalling the function $h_{n}$ appearing
there. Let $A\geq 1$ be a large real number, bounded in $n$, and set
$\mathsf{T}=An^{1/3}$. Sample $n-1$ non-intersecting Brownian bridges
$\breve{\bm{\mathcal{L}}}=(\breve{\mathcal{L}}_{1},\breve{\mathcal{L}}_{2},\ldots,\breve{\mathcal{L}}_{n-1})$
on $[-\mathsf{T},\mathsf{T}]$, starting at
$\bm{u}=\big{(}\mathcal{L}_{1}(-\mathsf{T}),\ldots,\mathcal{L}_{n-1}(-\mathsf{T})\big{)}$;
ending at
$\bm{v}=\big{(}\mathcal{L}_{1}(\mathsf{T}),\ldots,\mathcal{L}_{n-1}(\mathsf{T})\big{)}$;
and conditioned to lie above $h_{n}$. Since
$(\mathcal{L}_{1},\mathcal{L}_{2},\ldots,\mathcal{L}_{n-1})$ start at
$\bm{u}$, end at $\bm{v}$, and are conditioned to lie above $\mathcal{L}_{n}$,
(1.14) with height monotonicity (recall Section 1.3) yields a coupling between
$\bm{\mathcal{L}}$ and $\breve{\bm{\mathcal{L}}}$ such that
$\mathcal{L}_{j}=\breve{\mathcal{L}}_{j}+o(1)$ for each $j\in[1,n-1]$. See the
left side of Figure 1.9 for a depiction. Thus, we must lower bound the gaps of
$\breve{\bm{\mathcal{L}}}$.

To this end, let
$\bm{\mathsf{z}}=(\mathsf{z}_{1},\mathsf{z}_{2},\ldots,\mathsf{z}_{n-1})$
denote non-intersecting Brownian bridges on $[-\mathsf{T},\mathsf{T}]$,
starting and ending at $\bm{0}_{n-1}=(0,0,\ldots,0)$, and conditioned to lie
above a stretched semicircle
$f(s)=2^{1/2}\sigma\mathsf{T}(\mathsf{T}^{2}-s^{2})^{1/2}$, for some real
number $\sigma=1+o(1)$ near but slightly larger than $1$. The gaps of the
starting and ending data for $\bm{\mathsf{z}}$ are then smaller than those of
$\bm{u}$ and $\bm{v}$, and the lower boundary $h_{n}$ is more convex than $f$,
since $f^{\prime\prime}\leq-2^{-1/2}\sigma\leq-2^{-1/2}-o(1)\leq
h_{n}^{\prime\prime}$. Hence, gap monotonicity applies and implies that the
gaps of $\breve{\bm{\mathcal{L}}}$ stochastically dominate those of
$\bm{\mathsf{z}}$. We further show that top curves
$\big{(}\mathsf{z}_{1},\mathsf{z}_{2},\ldots)$ in $\bm{\mathsf{z}}$ converge
to an Airy line ensemble (8.1), by again using height monotonicity, now to
compare $\bm{\mathsf{z}}$ to the top $n-1$ curves of a Brownian watermelon
(with about $A^{3}n$ paths, the $n$-th of which is known to concentrate
tightly around the semicircle $f$). Therefore,
$\big{(}\mathsf{z}_{1}(t)-\mathsf{z}_{2}(t),\mathsf{z}_{2}(t)-\mathsf{z}_{3}(t),\ldots\big{)}$
converges to
$2^{-1/2}\cdot\big{(}\mathcal{R}_{1}(0)-\mathcal{R}_{2}(0),\mathcal{R}_{2}(0)-\mathcal{R}_{3}(0),\ldots\big{)}$.
Combining this with the above comparisons yields the Airy gap lower bound for
$\bm{\mathcal{L}}$. See the middle side of Figure 1.9 for a depiction.

Figure 1.9. Shown to the left and middle is a depiction for the proof of the
Airy gap lower bound for $\bm{\mathcal{L}}$. Shown to the right is a depiction
for the proof of the corresponding upper bound.

To prove the upper bound, we instead rely on the global law (1.12) (as opposed
to the curvature approximation (1.14)). Again let $n\gg 1$ be a large integer,
and now fix a small real number $\varepsilon\in(0,1)$, independent of $n$. Let
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{n})$ denote
Dyson Brownian motion, starting at time $-\varepsilon n^{1/3}$, with initial
data $\bm{u}^{\prime}=\big{(}\mathcal{L}_{1}(-\varepsilon
n^{1/3}),\ldots,\mathcal{L}_{n}(-\varepsilon n^{1/3})\big{)}$. Conditioning on
the locations
$\bm{v}^{\prime}=\big{(}\mathsf{y}_{1}(n^{20}),\ldots,\mathsf{y}_{n}(n^{20})\big{)}$
of $\bm{\mathsf{y}}$ at time $n^{20}$, the law of $\bm{\mathsf{y}}$ on
$[-\varepsilon n^{1/3},n^{20}]$ is then given by $n$ non-intersecting Brownian
bridges, starting at $\bm{u}^{\prime}$ and ending at $\bm{v}^{\prime}$,
without boundaries. One can verify (under a few mild modifications to the
above setup that we do not detail here) that the gaps
$\big{|}\mathsf{y}_{i}(n^{20})-\mathsf{y}_{j}(n^{20})\big{|}$ of
$\bm{\mathsf{y}}$ after being run for such a long time $n^{20}+\varepsilon
n^{1/3}$ are likely very large, and in particular greater than those
$\big{|}\mathcal{L}_{i}(n^{20})-\mathcal{L}_{j}(n^{20})\big{|}$ of
$\bm{\mathcal{L}}$ allowed by the gap upper bound (1.6). As $\bm{\mathsf{y}}$
has no lower boundary, gap monotonicity thus applies and implies that the gaps
of $\bm{\mathcal{L}}$ are stochastically dominated by those of
$\bm{\mathsf{y}}$. See the right side of Figure 1.9 for a depiction.

Results by Capitaine–Péché [28], on edge statistics of Dyson Brownian motion
under general initial data, can then be used on $\bm{\mathsf{y}}$. They
indicate that
$\varsigma\cdot\big{(}\mathsf{y}_{1}(t),\mathsf{y}_{2}(t),\ldots\big{)}$
converges (after recentering) to the Airy point process, where the rescaling
factor $\varsigma$ admits an explicit formula in terms of the initial data
$\bm{u}^{\prime}$. Using the approximation for $\bm{u}^{\prime}$ provided by
the global law (1.12), we show that $\varsigma\approx 2^{1/2}$ (8.4). It
follows that
$\big{(}\mathsf{y}_{1}(t)-\mathsf{y}_{2}(t),\mathsf{y}_{2}(t)-\mathsf{y}_{3}(t),\ldots\big{)}$
converges to
$2^{-1/2}\cdot\big{(}\mathcal{R}_{1}(0)-\mathcal{R}_{2}(0),\mathcal{R}_{2}(0)-\mathcal{R}_{3}(0),\ldots\big{)}$,
which with the above comparison between $\bm{\mathcal{L}}$ and
$\bm{\mathsf{y}}$ yields the Airy gap upper bound for $\bm{\mathcal{L}}$. See
Section 8 below for further details.

##### Airy Line Ensemble

By (1.2), the fact that the gaps of $\bm{\mathcal{L}}$ at any fixed time
$t\in\mathbb{R}$ are given by those of an Airy point process implies strong
concentration bounds for $\bm{\mathcal{L}}(t)$, up to an overall (random)
shift. In particular, fix large integers $N\gg n\gg 1$ and denote the
$N$-tuples
$\bm{u}=(u_{1},\ldots,u_{N})=\big{(}\mathcal{L}_{1}(-n^{1/3}),\ldots,\mathcal{L}_{N}(-n^{-1/3})\big{)}$
and
$\bm{v}=(v_{1},\ldots,v_{N})=\big{(}\mathcal{L}_{1}(n^{1/3}),\ldots,\mathcal{L}_{N}(n^{1/3})\big{)}$.
Then, (1.2) yields random variables $\mathfrak{u},\mathfrak{v}\in\mathbb{R}$
(we may take $\mathfrak{u}=u_{N}+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}N^{2/3}$
and $\mathfrak{v}=v_{N}+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}N^{2/3}$) such
that for $j\in[1,N]$ we have with high probability that

(1.16) $\displaystyle\begin{aligned}
&u_{j}=\mathfrak{u}-2^{-1/2}n^{2/3}-2^{-7/6}(3\pi)^{2/3}j^{2/3}+\mathcal{O}(j^{-1/3});\\\
&v_{j}=\mathfrak{v}-2^{-1/2}n^{2/3}-2^{-7/6}(3\pi)^{2/3}j^{-2/3}+\mathcal{O}(j^{-1/3}).\end{aligned}$

Condition on $\bm{\mathcal{L}}(-n^{1/3})$ and $\bm{\mathcal{L}}(n^{1/3})$
(thus fixing $\mathfrak{u}$ and $\mathfrak{v}$), so
$(\mathcal{L}_{1},\mathcal{L}_{2},\ldots,\mathcal{L}_{N})$ are $N$ non-
intersecting Brownian bridges starting at $\bm{u}$, ending at $\bm{v}$, and
conditioned to lie above $\mathcal{L}_{N+1}$. Further restrict to the (likely)
event that (1.16) holds. By subtracting an affine shift101010This is
ultimately what gives the residual independent affine shift in the
characterization 2.9 for $\bm{\mathcal{L}}$. from the $\mathcal{L}_{j}$ (given
in terms of $\mathfrak{u}$ and $\mathfrak{v}$ by
$t\cdot\xi+\zeta=t\cdot(\mathfrak{v}-\mathfrak{u})/2n^{1/3}+(\mathfrak{u}+\mathfrak{v})/2$),
we may assume $\mathfrak{u}=\mathfrak{v}=0$ in (1.16).

To show that $\bm{\mathcal{L}}$ is a parabolic Airy line ensemble, it then
suffices to establish the following more general statement (9.1). Sample $N$
non-intersecting Brownian bridges
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{N})$ on
$[-n^{1/3},n^{1/3}]$, starting at $\bm{u}$ and ending at $\bm{v}$ satisfying
(1.16) with $(\mathfrak{u},\mathfrak{v})=(0,0)$, and conditioned to lie above
a (not too irregular) lower boundary curve $f$. Then
$(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ converges to
$2^{-1/2}\cdot\bm{\mathcal{R}}$, as $n$ tends to $\infty$. To prove this, we
use (1.16) to sandwich $\bm{\mathsf{x}}$ between two parabolic Airy line
ensembles with approximately equal curvatures. This sort of idea was also
fruitful in analyzing edge statistics for random tilings [7], once a
concentration bound for the associated paths around explicit parabolas, as
strong as (1.3), was proven in the time direction.

In our context, we instead have the concentration bound (1.16) in the spatial
direction, so we must apply this idea in two ways. The first uses (1.16) to
sandwich $\bm{\mathsf{x}}$ between two parabolic Airy line ensembles on a tall
rectangle; see the left side of Figure 1.10 for a depiction. This enables us
to closely approximate $\mathsf{x}_{n}$ by a parabola in the time direction,
verifying that (1.3) holds for $\bm{\mathsf{x}}$ with that explicit choice of
$\mathfrak{f}_{n}$ (9.2). The second is to use this near-parabolicity to
sandwich $\bm{\mathsf{x}}$ between two parabolic Airy line ensembles on a long
interval (as in [7]), to conclude its convergence to the Airy line ensemble.
See the right side of Figure 1.10 for a depiction, and Section 9 below for
further details.

Figure 1.10. Shown to the left is the coupling on a short interval, used to
show that the curves in $\bm{\mathsf{x}}$ are close to parabolas (where the
top blue curve is at $\infty$ and thus not depicted). Shown to the right is
the coupling on a long interval, used to show convergence to the Airy line
ensemble.

####  1.7. Notation

In what follows, we set
$\overline{\mathbb{R}}=\mathbb{R}\cup\\{-\infty,\infty\\}$; we also let
$\mathbb{H}=\\{z\in\mathbb{C}:\operatorname{Im}z>0\\}$ denote the upper half-
plane, $\overline{\mathbb{H}}$ denote its closure,
$\mathbb{H}^{-}=\\{z\in\mathbb{C}:\operatorname{Im}z<0\\}$ denote the lower
half-plane; and $\overline{\mathbb{H}^{-}}$ denote its closure. Moreover, for
any subset $I\subseteq\mathbb{R}$ and measurable functions
$f,g:I\rightarrow\overline{\mathbb{R}}$ we write $f<g$ (equivalently, $g>f$)
if $f(t)<g(t)$ for each $t\in I$; we similarly write $f\leq g$ (equivalently,
$g\geq f$) if $f(t)\leq g(t)$ for each $t\in I$. For any sets $A_{0}\subseteq
A$ and function $f:A\rightarrow\mathbb{C}$, let $f|_{A_{0}}$ denote the
restriction of $f$ to $A_{0}$. In what follows, for any topological space $I$,
we let $\mathcal{C}(I)$ denote the space of real-valued, continuous functions
$f:I\rightarrow\mathbb{R}$.

For any integer $d\geq 1$ and $d$-tuple
$\gamma=(\gamma_{1},\gamma_{2},\ldots,\gamma_{d})\in\mathbb{Z}_{\geq 0}^{d}$,
define $|\gamma|=\sum_{j=1}^{d}\gamma_{j}$ and
$\partial_{\gamma}=\prod_{j=1}^{d}\partial_{j}$, where we have abbreviated the
differential operator $\partial_{j}=\partial_{x_{j}}=\partial/\partial x_{j}$
for each $j\in[1,d]$. For any integer $k\geq 0$ and open subset
$\mathfrak{R}\subseteq\mathbb{R}^{d}$, let $\mathcal{C}^{k}(\mathfrak{R})$
denote the set of $f\in\mathcal{C}(\overline{\mathfrak{R}})$ such that
$\partial_{\gamma}f$ is continuous on $\mathfrak{R}$, for each
$\gamma\in\mathbb{Z}_{\geq 0}^{d}$ with $|\gamma|\leq k$. Further let
$\mathcal{C}^{k}(\overline{\mathfrak{R}})$ denote the set of functions
$f\in\mathcal{C}^{k}(\mathfrak{R})$ such that $\partial_{\gamma}f$ extends
continuously to $\overline{\mathfrak{R}}$, for each $\gamma\in\mathbb{Z}_{\geq
0}^{d}$ with $|\gamma|\leq k$. For any function
$f\in\mathcal{C}(\mathfrak{R})$ and integer $k\in\mathbb{Z}_{\geq 0}$, we
further define the (semi)norms on $\|f\|_{0}=\|f\|_{0;\mathfrak{R}}$,
$[f]=[f]_{k;0;\mathfrak{R}}$, and
$\|f\|_{\mathcal{C}^{k}(\overline{\mathfrak{R}})}=\|f\|_{\mathcal{C}^{k}(\mathfrak{R})}=\|f\|_{k}=\|f\|_{k;0;\mathfrak{R}}$
on these spaces by

(1.17)
$\displaystyle\|f\|_{0}=\displaystyle\sup_{z\in\mathfrak{R}}\big{|}f(z)\big{|};\qquad[f]_{k}=\displaystyle\max_{\begin{subarray}{c}\gamma\in\mathbb{Z}_{\geq
0}^{d}\\\
|\gamma|=k\end{subarray}}\big{\|}\partial_{\gamma}f\|_{0};\qquad\|f\|_{\mathcal{C}^{k}(\overline{\mathfrak{R}})}=\displaystyle\sum_{j=0}^{k}[f]_{j}.$

Given an integer $d\geq 1$ and a subset $U\subset\mathbb{R}^{d}$, a function
$f:U\rightarrow\mathbb{C}$ is called real analytic if, for every point $z_{0}$
in the interior of $U$, it admits a power series expansion that converges
absolutely in a neighborhood of $z_{0}$.

For any real numbers $a,b\in\mathbb{R}$ with $a\leq b$, we set $\llbracket
a,b\rrbracket=[a,b]\cap\mathbb{Z}$. We also let $a\vee b=\max\\{a,b\\}$. For
any integer $k\geq 1$, we denote the entries of any $k$-tuple
$\bm{y}\in\mathbb{C}^{k}$ by $\bm{y}=(y_{1},y_{2},\ldots,y_{k})$, unless we
specify the indexing otherwise. For any $k$-tuples
$\bm{x},\bm{y}\in\mathbb{R}^{k}$, we write $\bm{x}<\bm{y}$ (equivalently,
$\bm{y}>\bm{x}$) if $x_{j}<y_{j}$ for each $j\in\llbracket 1,k\rrbracket$; we
similarly write $\bm{x}\leq\bm{y}$ (equivalently, $\bm{y}\geq\bm{x}$) if
$x_{j}\leq y_{j}$ for each $j\in\llbracket 1,k\rrbracket$. We also let
$\mathbb{W}_{k}=\\{\bm{y}\in\mathbb{R}^{k}:y_{1}>y_{2}>\cdots>y_{k}\\}$ and
let $\overline{\mathbb{W}}_{k}$ denote the closure of $\mathbb{W}_{k}$.
Further let $\bm{0}_{k}=(0,0,\ldots,0)\in\overline{\mathbb{W}}_{k}$, where $0$
appears with multiplicity $k$.

For any integer $k\geq 1$ and subset $\mathfrak{S}\subseteq\mathbb{R}^{k}$, we
let $\partial\mathfrak{S}$ denote the boundary of $\mathfrak{S}$; for any
point $z\in\mathbb{R}^{k}$, we also let
$\operatorname{dist}(z,\mathcal{S})=\inf_{s\in\mathcal{S}}|z-s|$. For any
complex numbers $a,b\in\mathbb{C}$, and vector $\bm{x}\in\mathbb{C}^{k}$, we
set $a\bm{x}+b=(ax_{1}+b,ax_{2}+b,\ldots,ax_{k}+b)\in\mathbb{C}^{k}$. For any
interval $I\subset\mathbb{R}^{k}$ and set $\mathcal{S}$ of vectors
$\mathcal{S}\subset\mathbb{R}^{k}$ or of functions
$\mathcal{S}\in\mathcal{C}(I)$, we similarly denote
$a\cdot\mathcal{S}+b=\\{as+b\\}_{s\in\mathcal{S}}$. For any additional such
set $\mathcal{S}^{\prime}$, denote
$\mathcal{S}+\mathcal{S}^{\prime}=\\{s+s^{\prime}:s\in\mathcal{S},s^{\prime}\in\mathcal{S}^{\prime}\\}$.

Let
$\mathscr{P}_{\operatorname{fin}}=\mathscr{P}_{\operatorname{fin}}(\mathbb{R})$
denote the set of nonnegative measures $\mu$ on $\mathbb{R}$ with finite total
mass, $\mu(\mathbb{R})<\infty$. Further let
$\mathscr{P}=\mathscr{P}(\mathbb{R})\subset\mathscr{P}_{\operatorname{fin}}$
denote the set of probability measures on $\mathbb{R}$, and let
$\mathscr{P}_{0}=\mathscr{P}_{0}(\mathbb{R})\subset\mathscr{P}$ denote the set
of probability measures that are compactly supported; the support of any
measure $\nu\in\mathscr{P}$ is denoted by $\operatorname{supp}\nu$. We say
that a probability measure $\mu\in\mathscr{P}$ has density $\varrho$ (with
respect to Lebesgue measure) if $\varrho:\mathbb{R}\rightarrow\mathbb{R}$ is a
measurable function satisfying $\mu(dx)=\varrho(x)dx$. For any real number
$x\in\mathbb{R}$, we let $\delta_{x}\in\mathscr{P}_{0}$ denote the delta
function at $x$. For any sequence
$\bm{a}=(a_{1},a_{2},\ldots,a_{n})\in\overline{\mathbb{W}}_{n}$, we denote its
_empirical measure_ $\operatorname{emp}(\bm{a})\in\mathscr{P}$ by

(1.18)
$\displaystyle\operatorname{emp}(\bm{a})=\displaystyle\frac{1}{n}\displaystyle\sum_{j=1}^{n}\delta_{a_{j}}.$

We denote the complement of any event $\mathscr{E}$ by
$\mathscr{E}^{\complement}$.

Throughout, given some integer parameter $n\geq 1$ and event $\mathscr{E}_{n}$
depending on $n$, we will often make statements of the following form. There
exists a constant $c>0$, independent of $n$ (but possibly dependent on other
parameters), such that $\mathbb{P}[\mathscr{E}_{n}^{\complement}]\leq f(c,n)$
holds for an explicit function $f:\mathbb{R}_{>0}\times\mathbb{Z}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$, which is non-decreasing in $c$ and
satisfies $\lim_{n\rightarrow\infty}f(c,n)=0$ and $\lim_{c\rightarrow
0}f(c,n)>1$ (an example is $f(c,n)=c^{-1}e^{-c(\log n)^{2}}$). When proving
such statements we will often implicitly (and without comment) assume that
$n\geq N_{0}$ is sufficiently large. Indeed, suppose there exist $N_{0}\geq 1$
and $c_{0}>0$ such that
$\mathbb{P}\big{[}\mathscr{E}_{n}^{\complement}\big{]}\leq f(c_{0},n)$ holds
whenever $n\geq N_{0}$. Since $\lim_{c\rightarrow 0}f(c,n)>1$, there exists a
constant $c_{1}>0$ such that for $n\leq N_{0}$ we have $f(c_{1},n)\geq 1$, in
which case $\mathbb{P}\big{[}\mathscr{E}_{n}^{\complement}\big{]}\leq 1\leq
f(c_{1},n)$ continues to hold. Thus, taking $c=\min\\{c_{0},c_{1}\\}$
guarantees that $\mathbb{P}\big{[}\mathscr{E}_{n}^{\complement}\big{]}\leq
f(c,n)$ holds for all $n\geq 1$.

#### Acknowledgements

Amol Aggarwal was partially supported by a Packard Fellowship for Science and
Engineering, a Clay Research Fellowship, by NSF grant DMS-1926686, and by the
IAS School of Mathematics. The research of Jiaoyang Huang is supported by NSF
grant DMS-2054835 and DMS-2331096. The authors heartily thank Alan Hammond for
extensive detailed and valuable comments throughout this paper. They further
thank Alexei Borodin and Ivan Corwin for providing very helpful suggestions,
and also Evgeni Dimitrov, Milind Hedge, Andrei Okounkov, Fabio Toninelli,
Horng-Tzer Yau and Lingfu Zhang for insightful conversations. They wish to
acknowledge the NSF grant DMS-1928930, which supported their participation in
the Fall 2021 semester program at MSRI in Berkeley, California titled,
“Universality and Integrability in Random Matrix Theory and Interacting
Particle Systems.”

###  2\. Results

####  2.1. Brownian Gibbs Property and the Airy Line Ensemble

In this section we introduce notation for non-intersecting Brownian bridges.
Let $\mathcal{S}\subseteq\mathbb{Z}_{\geq 1}$ and $I\subseteq\mathbb{R}$
denote intervals. Let $\mathfrak{X}=\mathfrak{X}(\mathcal{S};I)$ denote the
set of continuous functions $f:\mathcal{S}\times I\rightarrow\mathbb{R}$,
whose topology is determined by uniform convergence on compact subsets of
$\mathcal{S}\times I$; we denote the associated Borel $\sigma$-algebra by
$\mathscr{C}=\mathscr{C}(\mathcal{S}\times I)$. Since $\mathcal{S}$ is
discrete, any such $f$ can be interpreted as an element of
$\mathcal{S}\times\mathcal{C}(I)$. An _$\mathcal{S}\times I$ indexed line
ensemble_ is a $\mathfrak{X}$-valued random variable
$\bm{\mathsf{x}}\in\mathcal{S}\times\mathcal{C}(I)$ defined on a probability
space $(\Omega,\mathscr{B},\mathbb{P})$ that is
$(\mathscr{B},\mathscr{C})$-measurable. We will frequently denote such a line
ensemble by $\bm{\mathsf{x}}=(\mathsf{x}_{j})_{j\in\mathcal{S}}$, where
$\mathsf{x}_{j}:I\rightarrow\mathbb{R}$ is a (random) continuous function for
each $j\in\mathcal{S}$; in this case, we also set
$\bm{\mathsf{x}}(t)=\big{(}\mathsf{x}_{j}(t)\big{)}_{j\in\mathcal{S}}$ for
each $t\in I$.

We next provide notation for the probability measure of $n$ non-intersecting
Brownian bridges with given starting and ending points, and for given upper
and lower boundaries.

Figure 1.11. Depicted above is a sample from
$\mathfrak{Q}_{f;g}^{\bm{u};\bm{v}}(\sigma)$.

######  Definition 2.1.

Fix an integer $n\geq 1$; a real number $\sigma>0$; two $n$-tuples
$\bm{u},\bm{v}\in\overline{\mathbb{W}}_{n}$; an interval
$[a,b]\subseteq\mathbb{R}$; and measurable functions
$f,g:[a,b]\rightarrow\overline{\mathbb{R}}$ such that $f<g$, $f<\infty$, and
$g>-\infty$. Let $\mathsf{Q}_{f;g}^{\bm{u};\bm{v}}(\sigma)$ denote the law of
the $\llbracket 1,n\rrbracket\times[a,b]$ indexed line ensemble
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$, given by $n$ independent
Brownian motions of variance $\sigma$ on the time interval $t\in[a,b]$,
conditioned on satisfying the following three properties.

  1. (1)

The $\mathsf{x}_{j}$ do not intersect, that is,
$\bm{\mathsf{x}}(t)\in\mathbb{W}_{n}$ for each $t\in(a,b)$.

  2. (2)

The $\mathsf{x}_{j}$ start at $u_{j}$ and end at $v_{j}$, that is,
$\mathsf{x}_{j}(a)=u_{j}$ and $\mathsf{x}_{j}(b)=v_{j}$ for each
$j\in\llbracket 1,n\rrbracket$.

  3. (3)

The $\mathsf{x}_{j}$ are bounded above by $f$ and below by $g$, that is,
$f<\mathsf{x}_{j}<g$ for each $j\in\llbracket 1,n\rrbracket$.

We refer to $\bm{u}$ as _starting data_ for $\bm{\mathsf{x}}$, and to $\bm{v}$
as its _ending data_. We also refer to $f$ as the _lower boundary_ for
$\bm{\mathsf{x}}$, and to $g$ as its _upper boundary_. See Figure 1.11.

If $g=\infty$, then we abbreviate
$\mathsf{Q}_{f;g}^{\bm{u};\bm{v}}(\sigma)=\mathsf{Q}_{f}^{\bm{u};\bm{v}}(\sigma)$;
if additionally $f=-\infty$, we set
$\mathsf{Q}_{f;\infty}^{\bm{u};\bm{v}}(\sigma)=\mathsf{Q}_{f}^{\bm{u};\bm{v}}(\sigma)=\mathsf{Q}^{\bm{u};\bm{v}}(\sigma)$.
If $\sigma=1$, then we omit the parameter $\sigma$ from the notation, writing
$\mathsf{Q}_{f;g}^{\bm{u};\bm{v}}=\mathsf{Q}_{f;g}^{\bm{u};\bm{v}}(1)$,
$\mathsf{Q}_{f}^{\bm{u};\bm{v}}=\mathsf{Q}_{f}^{\bm{u};\bm{v}}(1)$, and
$\mathsf{Q}^{\bm{u};\bm{v}}=\mathsf{Q}^{\bm{u};\bm{v}}(1)$.

We next describe a resampling property from [39].

######  Definition 2.2.

Fix intervals $\mathcal{S}\subseteq\mathbb{Z}_{\geq 1}$ and
$I\subseteq\mathbb{R}$, as well as an $\mathcal{S}\times I$ indexed line
ensemble $\bm{\mathsf{x}}=(\mathsf{x}_{j})_{j\in\mathcal{S}}$. For any
integers $1\leq j\leq k$ such that $\llbracket
j,k\rrbracket\subseteq\mathcal{S}$, define the $\llbracket
j,k\rrbracket\times\mathbb{R}$ indexed line ensemble
$\bm{\mathsf{x}}_{\llbracket
j,k\rrbracket}=(\mathsf{x}_{j},\mathsf{x}_{j+1},\ldots,\mathsf{x}_{k})\in\llbracket
j,k\rrbracket\times\mathcal{C}(I)$. For any intervals
$\mathcal{S}^{\prime}\subseteq\mathcal{S}$ and $I^{\prime}\subseteq I$,
further let $\mathcal{F}_{\operatorname{ext}}(\mathcal{S}^{\prime}\times
I^{\prime})$ denote the $\sigma$-algebra generated by the
$\big{(}\mathsf{x}_{j}(t)\big{)}$, over all $j\notin\mathcal{S}^{\prime}$ or
$t\notin I^{\prime}$.

Fix a real number $\sigma>0$. We say that an $\mathcal{S}\times I$ indexed
line ensemble $\bm{\mathsf{x}}$ has the _Brownian Gibbs property_ of variance
$\sigma$ if we almost surely have
$\mathsf{x}_{1}(t)>\mathsf{x}_{2}(t)>\cdots$, for each real number
$t\in\mathbb{R}$, and the following holds, for any bounded intervals
$\llbracket k_{1},k_{2}\rrbracket\subseteq\mathcal{S}$ and $(a,b)\subseteq I$.
The law of $\big{(}\mathsf{x}_{j}(t)\big{)}$, over $(j,t)\in\llbracket
k_{1},k_{2}\rrbracket\times[a,b]$, conditional on
$\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
k_{1},k_{2}\rrbracket\times(a,b)\big{)}$ is given by the non-intersecting
Brownian bridge measure $\mathsf{Q}_{f,g}^{\bm{u};\bm{v}}(\sigma)$. Here, the
entrance and exit data $\bm{u},\bm{v}\in\mathbb{R}^{k_{2}-k_{1}+1}$ are given
by
$\bm{u}=\big{(}\mathsf{x}_{k_{1}}(a),\mathsf{x}_{k_{1}+1}(a),\ldots,\mathsf{x}_{k_{2}}(a)\big{)}$
and
$\bm{v}=\big{(}\mathsf{x}_{k_{1}}(b),\mathsf{x}_{k_{1}+1}(b),\ldots,\mathsf{x}_{k_{2}}(b)\big{)}$,
and the boundary data $f,g:[a,b]\rightarrow\mathbb{R}$ are given by
$f=\mathsf{x}_{k_{1}-1}|_{[a,b]}$, and $g=\mathsf{x}_{k_{2}+1}|_{[a,b]}$
(setting $\mathsf{x}_{j}=\infty$ if $j<\min\mathcal{S}$ and
$\mathsf{x}_{j}=-\infty$ if $j>\max\mathcal{S}$). If $\sigma=1$, we omit it
from the notation, saying that $\bm{\mathsf{x}}$ satisfies the Brownian Gibbs
property.

We next require some notation on edge statistics.

######  Definition 2.3.

For any $s,t,x,y\in\mathbb{R}$, the _extended Airy kernel_
$\mathcal{K}:\mathbb{R}^{4}\rightarrow\mathbb{R}$ is given by

$\displaystyle\mathcal{K}(s,x;t,y)=\displaystyle\int_{0}^{\infty}e^{u(t-s)}\operatorname{Ai}(x+u)\operatorname{Ai}(y+u)\mathrm{d}u,\qquad$
$\displaystyle\text{if $s\geq t$};$
$\displaystyle\mathcal{K}(s,x;t,y)=-\displaystyle\int_{-\infty}^{0}e^{u(t-s)}\operatorname{Ai}(x+u)\operatorname{Ai}(y+u)\mathrm{d}u,\qquad$
$\displaystyle\text{if $s<t$},$

where we recall that the Airy function
$\operatorname{Ai}:\mathbb{R}\rightarrow\mathbb{R}$ is given by

$\displaystyle\operatorname{Ai}(x)=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\infty}^{\infty}\cos\Big{(}\displaystyle\frac{z^{3}}{3}+xz\Big{)}\mathrm{d}z.$

From this, we define the Airy line ensemble.

######  Definition 2.4.

The _(stationary) Airy line ensemble_
$\bm{\mathcal{A}}=(\mathcal{A}_{1},\mathcal{A}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ is an infinite collection of random
continuous curves $\mathcal{A}_{i}:\mathbb{R}\rightarrow\mathbb{R}$, ordered
as $\mathcal{A}_{1}(t)>\mathcal{A}_{2}(t)>\cdots$ for each $t\in\mathbb{R}$,
such that

(2.1)
$\displaystyle\mathrm{d}\mathbb{P}\Bigg{[}\bigcap_{j=1}^{m}\big{\\{}(t_{j},y_{j})\in\bm{\mathcal{A}}\big{\\}}\Bigg{]}=\det\big{[}\mathcal{K}(t_{i},y_{i};t_{j},y_{j})\big{]}_{1\leq
i,j\leq m}\displaystyle\prod_{j=1}^{m}dy_{j},$

for any $(t_{1},y_{1}),(t_{2},y_{2}),\ldots,(t_{m},y_{m})\in\mathbb{R}^{2}$.
Here, we have written $(t,y)\in\bm{\mathcal{A}}$ if there exists some integer
$k\geq 1$ such that $\mathcal{A}_{k}(t)=y$. The existence of such an ensemble
was shown as [34, Theorem 3.1] (and the uniqueness follows from the
prescription (2.1) of its multi-point distributions). We abbreviate the
_parabolic Airy line ensemble_
$\bm{\mathcal{R}}=\big{(}\mathcal{A}_{1}(t)-t^{2},\mathcal{A}_{2}(t)-t^{2},\ldots\big{)}\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$, which may be viewed as a function
$\mathcal{R}:\mathbb{Z}_{\geq 1}\times\mathbb{R}\rightarrow\mathbb{R}$ by
setting $\mathcal{R}(i,t)=\mathcal{R}_{i}(t)=\mathcal{A}_{i}(t)-t^{2}$.

The following lemma from [39] states that the parabolic Airy line ensemble
satisfies the Brownian Gibbs property (after rescaling by $2^{-1/2}$).

######  Lemma 2.5 ([39, Theorem 3.1]).

The ensemble $2^{-1/2}\cdot\bm{\mathcal{R}}$ has the Brownian Gibbs property.

For any real number $s\in\mathbb{R}$, we let
$\Theta_{s}:\mathcal{C}(\mathbb{R})\rightarrow\mathcal{C}(\mathbb{R})$ denote
the translation operator acting on any function $f\in\mathcal{C}(\mathbb{R})$
by setting $\Theta_{s}f(x)=f(x+s)$, for each $x\in\mathbb{R}$. This operator
also acts on $\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line ensembles
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ by setting
$\Theta_{s}\bm{\mathsf{x}}=(\Theta_{s}\mathsf{x}_{1},\Theta_{s}\mathsf{x}_{2},\ldots)$.
As such, it also acts on measurable sets in the Borel $\sigma$-algebra
$\mathscr{C}=\mathscr{C}(\mathbb{Z}_{\geq 1}\times\mathbb{R})$. An
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line ensemble $\bm{\mathsf{x}}$
is called _translation-invariant_ if the law of $\bm{\mathsf{x}}$ is equal to
that of $\Theta_{s}\bm{\mathsf{x}}$, for each $s\in\mathbb{R}$.

We further say that an event $\mathscr{F}$ is translation-invariant if
$\Theta_{s}\mathscr{F}=\mathscr{F}$, for any $s\in\mathbb{R}$. For any real
number $\sigma>0$, we let $\operatorname{Tra}(\sigma)$ denote the set of
probability measures $\mu$ associated with a $\mathbb{Z}_{\geq
1}\times\mathbb{R}$ indexed line ensemble
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ satisfying the
Brownian Gibbs property, such that the ensemble
$\big{(}\mathsf{x}_{1}(t)+\sigma t^{2},\mathsf{x}_{2}(t)+\sigma
t^{2},\ldots\big{)}\in\mathbb{Z}_{\geq 1}\times\mathcal{C}(\mathbb{R})$ is
translation-invariant. We call a measure $\mu\in\operatorname{Tra}(\sigma)$
_extremal_ if, for any real number $p\in(0,1)$ and measures
$\mu_{1},\mu_{2}\in\operatorname{Tra}(\sigma)$ such that
$\mu=p\mu_{1}+(1-p)\mu_{2}$, we have $\mu_{1}=\mu=\mu_{2}$.

######  Lemma 2.6 ([34, Proposition 1.13]).

The law of $2^{-1/2}\cdot\bm{\mathcal{R}}$ is in
$\operatorname{Tra}(2^{-1/2})$ and is extremal.

####  2.2. Line Ensembles With Parabolic Decay

In this section we state our results, which constitute characterizations for
line ensembles satisfying the Brownian Gibbs property and certain growth
conditions. The latter conditions are explained through the following
definition and assumption, which describe the family of line ensembles we will
analyze. The definition introduces the event on which a given point in the top
curve of the ensemble is between two parabolas of approximately equal leading
coefficients (chosen to be $-2^{-1/2}$, to agree with the behavior of
$2^{-1/2}\cdot\bm{\mathcal{R}}$); the assumption states that this event is
likely.

######  Definition 2.7.

Let $\mathcal{S}\subseteq\mathbb{Z}_{\geq 1}$ denote an interval with
$1\in\mathcal{S}$; let $I\subseteq\mathbb{R}$ denote an interval (not
necessarily bounded); and let
$\bm{\mathsf{x}}=(\mathsf{x}_{s})_{s\in\mathcal{S}}\in\mathcal{S}\times\mathcal{C}(I)$
denote an $\mathcal{S}\times I$ indexed line ensemble. For any real numbers
$\varepsilon>0$, $C>1$ and $t\in I$, define the event
$\textbf{PAR}_{\varepsilon}(t;C)=\textbf{PAR}_{\varepsilon}^{\bm{\mathsf{x}}}(t;C)$
by

(2.2)
$\displaystyle\textbf{PAR}_{\varepsilon}(t;C)=\big{\\{}-(2^{-1/2}+\varepsilon)t^{2}-C\leq\mathsf{x}_{1}(t)\leq-(2^{-1/2}-\varepsilon)t^{2}+C\big{\\}}.$

######  Assumption 2.8.

Let
$\bm{\mathcal{L}}=(\mathcal{L}_{1},\mathcal{L}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ denote a $\mathbb{Z}_{\geq
1}\times\mathbb{R}$ indexed line ensemble satisfying the Brownian Gibbs
property. Assume that there exists a function111111Whenever adopting this
assumption, we will view $\mathfrak{K}$ as fixed. In particular, underlying
constants might depend on $\mathfrak{K}$, even when this dependence is not
stated explicitly. $\mathfrak{K}:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$
such that, for each $\varepsilon>0$, we have
$\mathbb{P}\big{[}\textbf{PAR}_{\varepsilon}^{\bm{\mathcal{L}}}(t;C)\big{]}\geq
1-\varepsilon$, for any real numbers $t\in\mathbb{R}$ and
$C\geq\mathfrak{K}(\varepsilon)$.

The following assumption classifies those line ensembles satisfying (2.8) as a
combination of rescaled parabolic Airy line ensembles; it will be established
in Section 3.5 below.

######  Theorem 2.9.

Adopt 2.8. There exist two random variables
$\mathfrak{l},\mathfrak{c}\in\mathbb{R}$, and a parabolic Airy line ensemble
$\bm{\mathcal{R}}=(\mathcal{R}_{1},\mathcal{R}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ (as in Definition 2.4) independent from them,
such that
$\mathcal{L}_{j}(t)=2^{-1/2}\cdot\mathcal{R}_{j}(t)+\mathfrak{l}t+\mathfrak{c}$,
for each $(j,t)\in\mathbb{Z}_{\geq 1}\times\mathbb{R}$.

Let us discuss a few consequences of 2.9. To do so, it will be useful to scale
the parabolic Airy line ensemble. For any real number $\sigma>0$, define the
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line ensemble
$\bm{\mathcal{R}}^{(\sigma)}=\big{(}\mathcal{R}_{1}^{(\sigma)},\mathcal{R}_{2}^{(\sigma)},\ldots\big{)}\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ by setting

(2.3)
$\displaystyle\mathcal{R}_{j}^{(\sigma)}(t)=\sigma^{-1}\cdot\mathcal{R}_{j}(\sigma^{2}t),\qquad\text{for
each $(j,t)\in\mathbb{Z}_{\geq 1}\times\mathbb{R}$}.$

######  Remark 2.10.

Since, for any real number $\sigma>0$, the law of any Brownian bridge $B(t)$
equal to that of $\sigma^{-1}\cdot B(\sigma^{2}t)$, and since
$2^{-1/2}\cdot\bm{\mathcal{R}}$ satisfies the Brownian Gibbs property (recall
2.5), $2^{-1/2}\cdot\bm{\mathcal{R}}^{(\sigma)}$ does as well for any
$\sigma>0$.

From 2.9, we can quickly derive the following corollary classifying line
ensembles with more specific rates of decay; it will be established in Section
3.7 below.

######  Corollary 2.11.

Fix a real number $\sigma>0$ and a $\mathbb{Z}_{\geq 1}\times\mathbb{R}$
indexed line ensemble
$\bm{\mathcal{L}}=(\mathcal{L}_{1},\mathcal{L}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ satisfying the Brownian Gibbs property; set
$q=2^{1/6}\sigma^{1/3}$.

  1. (1)

Assume for any real number $\varepsilon>0$ that there exists a constant
$C=C(\varepsilon)>1$ such that

(2.4)
$\displaystyle\mathbb{P}\big{[}-\sigma(1+\varepsilon)t^{2}-C\leq\mathcal{L}_{1}(t)\leq-\sigma(1-\varepsilon)t^{2}+C\big{]}\geq
1-\varepsilon,$

for any real number $t\in\mathbb{R}$. Then there exist two random variables
$\mathfrak{l},\mathfrak{c}\in\mathbb{R}$ and a scaled parabolic Airy line
ensemble
$\bm{\mathcal{R}}^{(q)}=\big{(}\mathcal{R}_{1}^{(q)},\mathcal{R}_{2}^{(q)},\ldots\big{)}\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ (as in (2.3)) independent from them, such
that
$\mathcal{L}_{j}(t)=2^{-1/2}\cdot\mathcal{R}_{j}^{(q)}(t)+\mathfrak{l}t+\mathfrak{c}$,
for each $(j,t)\in\mathbb{Z}_{\geq 1}\times\mathbb{R}$.

  2. (2)

Further fix a real number $\ell\in\mathbb{R}$. Assume for any real number
$\varepsilon>0$ that there exists a constant $C=C(\varepsilon)>1$ such that

(2.5) $\displaystyle\mathbb{P}\big{[}-\sigma t^{2}+\ell
t-\varepsilon|t|-C\leq\mathcal{L}_{1}(t)\leq-\sigma t^{2}+\ell
t+\varepsilon|t|+C\big{]}\geq 1-\varepsilon,$

for any real number $t\in\mathbb{R}$. Then there exists a random variable
$\mathfrak{c}\in\mathbb{R}$ and a scaled parabolic Airy line ensemble
$\bm{\mathcal{R}}^{(q)}=\big{(}\mathcal{R}_{1}^{(q)},\mathcal{R}_{2}^{(q)},\ldots\big{)}\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ (as in (2.3)) independent from
$\mathfrak{c}$, such that
$\mathcal{L}_{j}(t)=2^{-1/2}\cdot\mathcal{R}_{j}^{(q)}(t)+\ell
t+\mathfrak{c}$, for each $(j,t)\in\mathbb{Z}_{\geq 1}\times\mathbb{R}$.

From 2.11, we can quickly establish the following result characterizing
extremal line ensembles satisfying the Brownian Gibbs property; it will also
be established in Section 3.7 below.

######  Corollary 2.12.

Let
$\bm{\mathcal{L}}=(\mathcal{L}_{1},\mathcal{L}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ be a $\mathbb{Z}_{\geq 1}\times\mathbb{R}$
indexed line ensemble; denote its associated probability measure by $\mu$. If
$\mu\in\operatorname{Tra}(2^{-1/2})$ and $\mu$ is extremal, then there exists
a (deterministic) constant $c\in\mathbb{R}$ such that
$\mathcal{L}_{j}(t)=2^{-1/2}\cdot\mathcal{R}_{j}(t)+c$, for each
$(j,t)\in\mathbb{Z}_{\geq 1}\times\mathbb{R}$.

###  3\. Proof of Characterization

In this section we establish 2.9 assuming several statements that will be
established later, which consist of two types of results. The first provides
various properties of the line ensemble $\bm{\mathcal{L}}$ (defined on the
infinite line $\mathbb{R}$) satisfying 2.8; they are given in Section 3.1 and
Section 3.2. The second analyzes the asymptotic behaviors of families of non-
intersecting Brownian bridges on finite intervals; they are given in Section
3.3 and Section 3.4. We then establish 2.9 in Section 3.5 and Section 3.6; we
conclude by establishing 2.11 and 2.12 as consequences of 2.9 in Section 3.7.

####  3.1. On-Scale Events

In this section we state two results indicating a coarse similarity between
any line ensemble $\bm{\mathcal{L}}$ satisfying 2.8 and the rescaled parabolic
Airy line ensemble $2^{-1/2}\cdot\bm{\mathcal{R}}$ of Definition 2.4. The
first will imply that the top curve of $\bm{\mathcal{L}}$ is close to (within
$o(n^{2/3})$ of) a parabola along a long interval (of length growing faster
than $n^{2/3}$). The second will bound the locations of and gaps between (and
also the Hölder regularity of) the paths in $\bm{\mathcal{L}}$, showing that
they are of the same order as those in $2^{-1/2}\cdot\bm{\mathcal{R}}$.

To make these notions precise, we begin with the following two definitions.
The first prescribes a certain mesh $\mathcal{T}_{k}$, and also the event on
which PAR (from Definition 2.7) holds at each point on an interval. The second
prescribes the event on which the $k$-th curve of a line ensemble is above and
below the parabola $-2^{-1/2}t^{2}$, within some prescribed errors ($b$ and
$B$). Throughout this section, we let
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ denote a $\mathbb{Z}_{\geq
1}\times\mathbb{R}$ indexed line ensemble satisfying the Brownian Gibbs
property; the results in this section, except for 6.1 below, will apply to a
general such ensemble $\bm{\mathsf{x}}$ (not necessarily subject to 2.8).

######  Definition 3.1.

For any integer $k\geq 1$; real numbers $\alpha\in(0,1)$, $\varepsilon,C>0$,
and $A\geq 1$; and subset $\mathcal{T}\subset\mathbb{R}$, define the event
$\textbf{PAR}_{\varepsilon}(\mathcal{T};C)=\textbf{PAR}_{\varepsilon}^{\bm{\mathsf{x}}}(\mathcal{T};C)$
and then set $\mathcal{T}_{k}(\alpha;A)\subset\mathbb{R}$ by

$\displaystyle\textbf{PAR}_{\varepsilon}(\mathcal{T};C)=\bigcap_{t\in\mathcal{T}}\textbf{PAR}_{\varepsilon}^{\bm{\mathsf{x}}}(t;C);\qquad\mathcal{T}_{k}(\alpha;A)=\big{\\{}x\in[-Ak^{1/3},Ak^{1/3}]:x\in(\alpha
k^{1/3})\mathbb{Z}\big{\\}}.$

######  Definition 3.2.

For any integer $k\geq 1$ and real numbers $t,b,B\in\mathbb{R}$ with $B\geq
b$; define the _medium position events_
$\textbf{MED}_{k}(t;b;B)=\textbf{MED}_{k}^{\bm{\mathsf{x}}}(t;b;B)$ and
$\textbf{MED}_{k}(t;B)=\textbf{MED}_{k}^{\bm{\mathsf{x}}}(t;B)$ by

$\displaystyle\quad\textbf{MED}_{k}(t;b;B)=\big{\\{}-B\leq\mathsf{x}_{k}(t)+2^{-1/2}t^{2}\leq-b\big{\\}};\qquad\textbf{MED}_{k}(t;B)=\textbf{MED}_{k}(t;-B;B).$

Moreover, for any subset $\mathcal{T}\subseteq\mathbb{R}$, define (as in
Definition 3.1) the events
$\textbf{MED}_{k}(\mathcal{T};b;B)=\textbf{MED}_{k}^{\bm{\mathsf{x}}}(\mathcal{T};b;B)$
and
$\textbf{MED}_{k}(\mathcal{T};B)=\textbf{MED}_{k}^{\bm{\mathsf{x}}}(\mathcal{T};B)$
by

$\displaystyle\textbf{MED}_{k}(\mathcal{T};b;B)=\bigcap_{t\in\mathcal{T}}\textbf{MED}_{k}(t;b;B);\qquad\textbf{MED}_{k}(\mathcal{T};B)=\bigcap_{t\in\mathcal{T}}\textbf{MED}_{k}(t;B).$

For each of these events, if $k=1$, then we abbreviate the _top curve event_
$\textbf{TOP}=\textbf{MED}_{1}$. Observe in particular that
$\textbf{TOP}(t;\varepsilon t^{2}+C)=\textbf{PAR}_{\varepsilon}(t;C)$ for any
real numbers $t\in\mathbb{R}$ and $\varepsilon,C>0$.

The following proposition states that, if $\textbf{PAR}_{\varepsilon}$ (with
very small $\varepsilon$) holds at a sufficiently fine mesh of points, then
the top curve $\mathsf{x}_{1}$ of $\bm{\mathsf{x}}$ is likely within
$o(n^{1/3})$ of the parabola $2^{-1/2}t^{2}$, at every point on an interval of
length larger than $n^{2/3}$. It is established in Section 6.1 below. The
following corollary applies 3.3 to a line ensemble $\bm{\mathcal{L}}$
satifsying 2.8.

######  Proposition 3.3.

There exists a constant $C>1$ such that the following holds. Fix real numbers
$\alpha,\varepsilon,\omega\in(0,1/4)$ and $A\geq 1$ such that

$\displaystyle\vartheta=\vartheta(\alpha,\varepsilon,\omega)=7500A^{2}(\alpha+\varepsilon+\omega)\leq
2^{-50}.$

For any integer $k>A\alpha^{-4}$, we have

$\displaystyle\mathbb{P}\Big{[}\textbf{\emph{PAR}}_{\varepsilon}\big{(}\mathcal{T}_{k}(\alpha;15A);\omega
k^{2/3}\big{)}\cap\textbf{\emph{TOP}}\big{(}[-10Ak^{1/3},10Ak^{1/3}];\vartheta
k^{2/3}\big{)}^{\complement}\Big{]}\leq Ce^{-(\log k)^{2}}.$

######  Corollary 3.4.

Adopt 2.8 and fix real numbers $B\geq 1$ and $\delta,\vartheta>0$. There
exists a constant $C=C(B,\delta,\vartheta)>1$ such that, for $n\geq C$, we
have

$\displaystyle\mathbb{P}\Big{[}\textbf{\emph{TOP}}^{\bm{\mathcal{L}}}\big{(}[-Bn^{1/3},Bn^{1/3}];\vartheta
n^{2/3}\big{)}\Big{]}\geq 1-\delta.$

###### Proof.

We may assume in what follows that $B\geq 10$ and that $\vartheta\leq
2^{-50}$, due to the inclusion
$\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-Bn^{1/3},Bn^{1/3}];\vartheta
n^{2/3}\big{)}\subseteq\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-B^{\prime}n^{1/3},B^{\prime}n^{1/3}];\vartheta^{\prime}n^{2/3}\big{)}$
whenever $B\geq B^{\prime}$ and $\vartheta\leq\vartheta^{\prime}$. Define real
numbers $A\geq 1$ and $\alpha,\varepsilon,\omega>0$ (all implicitly dependent
on $B$ and $\vartheta$) by

(3.1) $\displaystyle
A=\displaystyle\frac{B}{10},\qquad\text{and}\qquad\alpha=\omega=\varepsilon=\displaystyle\frac{\vartheta}{22500A^{2}},\qquad\text{so
that}\qquad\vartheta=7500A^{2}(\alpha+\varepsilon+\omega).$

Then 2.8 implies for sufficiently large $n$ that

$\displaystyle\displaystyle\inf_{|t|\leq
15An^{1/3}}\mathbb{P}\big{[}\textbf{PAR}_{\varepsilon}^{\bm{\mathcal{L}}}(t;\omega
n^{2/3})\big{]}\geq 1-\displaystyle\frac{\alpha\delta}{90A}.$

This, a union bound, and the facts that
$\mathcal{T}_{n}(\alpha;15A)\subset[-15An^{1/3},15An^{1/3}]$ and
$\big{|}\mathcal{T}_{n}(\alpha;15A)\big{|}\leq 45A\alpha^{-1}$ together imply
that

(3.2)
$\displaystyle\mathbb{P}\Big{[}\textbf{PAR}_{\varepsilon}^{\bm{\mathcal{L}}}\big{(}\mathcal{T}_{n}(\alpha;15A);\omega
n^{2/3}\big{)}\Big{]}\geq 1-\displaystyle\frac{\delta}{2}.$

It follows that there exists a constant $C_{1}>1$ such that

$\displaystyle\mathbb{P}\Big{[}\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[$
$\displaystyle-Bn^{1/3},Bn^{1/3}];\vartheta n^{2/3}\big{)}\Big{]}$
$\displaystyle=\mathbb{P}\Big{[}\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-10An^{1/3},10An^{1/3}];\vartheta
n^{2/3}\big{)}\Big{]}\geq 1-\displaystyle\frac{\delta}{2}-C_{1}e^{-(\log
n)^{2}},$

where in the first statement we used (3.1), and in the second we applied
(3.2), 3.3, and a union bound; taking $n$ sufficiently large then yields the
corollary. ∎

We next define two additional events. The first states an upper bound for the
gaps between the paths in $\bm{\mathsf{x}}$, indicating that they are
comparable to those of the Airy line ensemble (in which the distance between
the $i$-th and $j$-th curves is of order about $|j^{2/3}-i^{2/3}|$, which can
be deduced from 4.34 below). The second provides a Hölder type estimate for
the paths in $\bm{\mathsf{x}}$ (that is fairly weak in comparison to the one
that holds for the Airy line ensemble).

######  Definition 3.5.

For any integer $k\geq 1$; real number $B\in\mathbb{R}$; and subset
$\mathcal{T}\subset\mathbb{R}$, define the _gap event_
$\textbf{GAP}_{k}(\mathcal{T};B)=\textbf{GAP}_{k}^{\bm{\mathsf{x}}}(\mathcal{T};B)$
by

(3.3)
$\displaystyle\textbf{GAP}_{k}\big{(}\mathcal{T};B)=\bigcap_{t\in\mathcal{T}}\bigcap_{1\leq
i<j\leq k}\big{\\{}\mathsf{x}_{i}(t)-\mathsf{x}_{j}(t)\leq
B(j^{2/3}-i^{2/3})+(\log k)^{25}i^{-1/3}\big{\\}}.$

######  Definition 3.6.

For any integers $k,n\geq 1$; real numbers $B,\varsigma\geq 0$; and subset
$\mathcal{T}\subset\mathbb{R}$, define the _Hölder regular event_
$\textbf{REG}_{k}(\mathcal{T};B;\varsigma;n)=\textbf{REG}_{k}^{\bm{\mathsf{x}}}(\mathcal{T};B;\varsigma;n)$
by

(3.4)
$\displaystyle\textbf{REG}_{k}\big{(}\mathcal{T};B;\varsigma;n)=\bigcap_{t,t+s\in\mathcal{T}}\Big{\\{}\big{|}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)\big{|}\leq
4\big{(}n|t-s|\big{)}^{1/2}+B|s|+\varsigma\Big{\\}}.$

We next define an event that is formed from intersecting the ones above; it
prescribes when the gaps and locations of $\bm{\mathsf{x}}$ are “on-scale”
with respect to (that is, within constant factors of) those in the parabolic
Airy line ensemble (in addition to imposing the Hölder type regularity of
Definition 3.6). In what follows, if one examines curves $\mathsf{x}_{k}$ with
$k$ of order $n$, then the relevant scales of the time $t$ and space
$\mathsf{x}$ parameters are $n^{1/3}$ and $n^{2/3}$, respectively; see Figure
1.2.

######  Definition 3.7.

For any integer $n\geq 1$ and real numbers $A,B,D,R>0$, define the _on-scale
event_ $\textbf{SCL}_{n}(A;B;D;R)=\textbf{SCL}_{n}^{\bm{\mathsf{x}}}(A;B;D;R)$
by

$\displaystyle\textbf{SCL}_{n}(A;B;D;R)$ $\displaystyle=\bigcap_{k=\lceil
n/B\rceil}^{\lfloor
Bn\rfloor}\textbf{MED}_{k}\Big{(}[-3An^{1/3},3An^{1/3}];\displaystyle\frac{k^{2/3}}{15000};1500k^{2/3}\Big{)}$
$\displaystyle\qquad\cap\bigcap_{k=\lceil n/B\rceil}^{\lfloor
Bn\rfloor}\textbf{REG}_{k}\big{(}[-An^{1/3},An^{1/3}];4AB^{1/3};n^{-D};Bn\big{)}$
$\displaystyle\qquad\cap\textbf{GAP}_{n}\big{(}[-An^{1/3},An^{1/3}];R).$

The next theorem indicates that, if the top curve of $\bm{\mathsf{x}}$ is
close to a parabola on a long interval, then the on-scale event likely holds
on another long (but slightly shorter) interval. It is proven in Section 7.1
below.

######  Theorem 3.8.

For any real numbers $A,B,D,R\geq 2$, there exist constants $c=c(A,B,D)>0$,
$C_{1}=C_{1}(B)>1$, and $C_{2}=C_{2}(A,B)>1$ such that the following holds. If
$R\geq C_{2}$, then

$\displaystyle\mathbb{P}\Big{[}\textbf{\emph{TOP}}\big{(}[-C_{2}n^{1/3},C_{2}n^{1/3}];C_{1}^{-1}n^{2/3}\big{)}\cap\textbf{\emph{SCL}}_{n}(A;B;D;R)^{\complement}\Big{]}\leq
c^{-1}e^{-c(\log n)^{2}}.$

####  3.2. Global Law and Regular Profile Events

Recall that 3.8 indicates when paths in a line ensemble are within constant
factors of those $\mathcal{R}_{j}(t)$ in the parabolic Airy line ensemble
$\bm{\mathcal{R}}$. Those of the latter are known to concentrate around a
deterministic profile. Throughout, we define the function
$\mathfrak{G}:\mathbb{R}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ by
setting

(3.5)
$\displaystyle\mathfrak{G}(t,x)=-2^{-1/2}t^{2}-2^{-7/6}(3\pi)^{2/3}x^{2/3},\qquad\text{for
any $(t,x)\in\mathbb{R}\times\mathbb{R}_{\geq 0}$}.$

Then we have (see 4.34 or 10.13 below) with high probability that

(3.6) $\displaystyle
2^{-1/2}\cdot\mathcal{R}_{j}(t)\approx\mathfrak{G}(t,j)+\mathcal{O}(j^{-1/3})=n^{2/3}\cdot\mathfrak{G}(tn^{-1/3},jn^{-1})+\mathcal{O}(j^{-1/3}),$

for any integer $n\geq 1$. We first state a result indicating that the curves
$\mathcal{L}_{j}(t)$ of an ensemble $\bm{\mathcal{L}}$ under 2.8 satisfy the
bound
$\mathcal{L}_{j}(t)=n^{2/3}\cdot\mathfrak{G}(tn^{-1/3},jn^{-1})+o(n^{2/3})$.
This might be viewed as a global law, or limit shape, for the line ensemble
$\bm{\mathcal{L}}$. It is weaker than (3.6) but improves on the MED event part
appearing in SCL (recall Definition 3.7) arising in 3.8.

We begin with the following definition for the event on which the global law
holds.

######  Definition 3.9.

Fix an infinite sequence
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ of continuous functions. For any integer
$n\geq 1$, and real numbers $\delta,B>0$, define the _global law event_
$\textbf{GBL}_{n}(\delta;B)=\textbf{GBL}_{n}^{\bm{\mathsf{x}}}(\delta;B)$ by
(recalling (3.5))

$\displaystyle\textbf{GBL}_{n}^{\bm{\mathsf{x}}}(\delta;B)=\bigcap_{|t|\leq
Bn^{1/3}}\bigcap_{j=1}^{\lfloor
Bn\rfloor}\Big{\\{}\big{|}\mathsf{x}_{j}(t)-n^{2/3}\cdot\mathfrak{G}(tn^{-1/3},jn^{-1})\big{|}\leq\delta
n^{2/3}\Big{\\}}.$

The following theorem, to be established in Section 20.1 below, states that
the global law event likely holds for the ensemble $\bm{\mathcal{L}}$ from
2.8.

######  Theorem 3.10.

Adopt 2.8, and fix real numbers $B>1$ and $\delta>0$. There exists a constant
$C=C(B,\delta)>0$ such that, for $n\geq C$, we have

$\displaystyle\mathbb{P}\big{[}\textbf{\emph{GBL}}_{n}^{\bm{\mathcal{L}}}(\delta;B)\big{]}\geq
1-\delta.$

We next state results indicating that the locations of the paths in a line
ensemble $\bm{\mathcal{L}}$ satisfying 2.8 approximate a “regular profile.”
The following definition makes that notion more precise.

######  Definition 3.11.

Fix real numbers $a<b$, and let
$\bm{x}=(x_{1},x_{2},\ldots,x_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$ denote a sequence of
functions. For any real numbers $\delta,B\geq 0$ and $t\in[a,b]$, we define
the _regular profile_ event $\textbf{PFL}^{\bm{x}}(t;\delta;B)$ to be that on
which there exists a function $\gamma_{t}:[0,1]\rightarrow\mathbb{R}$ such
that

(3.7) $\displaystyle\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}x_{j}(t)-\gamma_{t}(jn^{-1})\big{|}\leq\delta,\qquad\text{and}\qquad\big{\|}\gamma_{t}-\gamma_{t}(0)\big{\|}_{\mathcal{C}^{50}}\leq
B.$

The first bound in (3.7) states that $\bm{x}$ approximates $\gamma_{t}$ at
time $t$; the second states that $\gamma_{t}$ is regular.

We will show through the following theorem that the
$\\{n+1,n+2,\ldots,2n\\}$-th curves of the line ensemble $\bm{\mathcal{L}}$
from 2.8 satisfy the regular profile event with high probability, after
rescaling and restricting to an intersection of TOP events (recall Definition
3.2). It will be established in Section 19.2 below.

######  Theorem 3.12.

Adopt 2.8, and fix a real number $A>1$. There exist constants
$\omega\in(0,1/2)$, $c=c(A)\in(0,1)$, and $C=C(A)>1$ such that the following
holds. Let $n\geq 1$ be an integer, and define the ensemble
$\bm{l}=\bm{l}^{(n)}=(l_{1},l_{2},\ldots,l_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}(\mathbb{R})$ by setting
$l_{j}(t)=n^{-2/3}\cdot\mathcal{L}_{j+n}(tn^{1/3})$, for each
$(j,t)\in\llbracket 1,n\rrbracket\times\mathbb{R}$. Then, we have

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{|t|\leq
An^{1/3}}\textbf{\emph{PFL}}^{\bm{l}}\big{(}t;n^{-1}(\log
n)^{6};C\big{)}^{\complement}\cap\bigcap_{k=1}^{\lfloor\omega^{-2}\rfloor}\textbf{\emph{TOP}}^{\bm{\mathcal{L}}}\big{(}[-Cn^{k\omega/3},Cn^{k\omega/3}$
$\displaystyle];cn^{2k\omega/3})\Bigg{]}\leq Cn^{-50}.$

####  3.3. Second Derivative Estimates for Paths

In this section we state 11.1 below. It indicates that, under certain
conditions, the paths in a family of non-intersecting Brownian bridges are
close to (random) curves with nearly constant second derivatives. The
following assumption more precisely prescribes these conditions on the
bridges, which will take place on the time interval $[-\xi n^{1/3},\xi
n^{1/3}]$ for some bounded $\xi>0$. The first condition (3.8) below states
that the boundary data (consisting of the entrance and exit data, as well as
the lower and upper boundaries) are approximated by a function $G$. The second
(3.10) states that the bridges likely satisfy a regular profile event at any
fixed time $s\in[-\xi n^{1/3},\xi n^{1/3}]$.

######  Assumption 3.13.

Fix real numbers $\delta\in(0,1/2)$ and $B>1$. Further let $\mathsf{T}>1$ and
$\xi\in(B^{-1},B)$ be real numbers such that $\mathsf{T}=n^{1/3}\xi$. Set
$\mathfrak{R}=[-\xi,\xi]\times[0,1]$, and let
$G:\overline{\mathfrak{R}}\rightarrow\mathbb{R}$ denote a continuous function.
Fix two $n$-tuples $\bm{u},\bm{v}\in\overline{\mathbb{W}}_{n}$ and two
functions $f,g:[-\mathsf{T},\mathsf{T}]\rightarrow\mathbb{R}$, such that $f<g$
and $f(-\mathsf{T})<u_{n}<u_{1}<g(-\mathsf{T})$ and
$f(\mathsf{T})<v_{n}<v_{1}<g(\mathsf{T})$. Suppose that

(3.8) $\displaystyle\begin{aligned} &\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}n^{-2/3}u_{j}-G(-\xi,jn^{-1})\big{|}\leq\delta;\qquad\qquad\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}n^{-2/3}v_{j}-G(\xi,jn^{-1})\big{|}\leq\delta;\\\
&\displaystyle\sup_{s\in[-\mathsf{T},\mathsf{T}]}\big{|}n^{-2/3}f(s)-G(n^{-1/3}s,1)\big{|}<\delta;\qquad\displaystyle\sup_{s\in[-\mathsf{T},\mathsf{T}]}\big{|}n^{-2/3}g(s)-G(n^{-1/3}s,0)\big{|}<\delta.\end{aligned}$

Let
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T},\mathsf{T}]\big{)}$ be a
family of $n$ non-intersecting Brownian bridges sampled from the measure
$\mathsf{Q}_{f;g}^{\bm{u};\bm{v}}$. Further define the rescaled family of non-
intersecting Brownian bridges $\bm{x}=(x_{1},x_{2},\ldots,x_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\xi,\xi]\big{)}$ by setting

(3.9) $\displaystyle
x_{j}(s)=n^{-2/3}\cdot\mathsf{x}_{j}(n^{1/3}s),\qquad\text{for each
$(j,s)\in\llbracket 1,n\rrbracket\times[-\xi,\xi]$},$

and assume for each real number $s\in[-\xi,\xi]$ that

(3.10)
$\displaystyle\mathbb{P}\big{[}\textbf{PFL}^{\bm{x}}(s;n^{-19/20};B)\big{]}\geq
1-n^{-20}.$

Observe that the event in (3.10) depends not only on the boundary data, but
also on the random bridges in $\bm{\mathsf{x}}$ themselves. It imposes that we
somehow knew “in advance” that these bridges likely have some regularity. In
our eventual context, this knowledge will come from 3.12.

The following theorem, to be established in Section 11.1 below, considers
Brownian bridges under 3.13 with the specific choice of $G(t,x)$ given (up to
a shift in its arguments) by $\mathfrak{G}$ of (3.5); it has constant second
derivative $-2^{-1/2}$ in $t$. It then states that the paths in
$\bm{\mathsf{x}}$ are near curves that also have nearly constant second
derivative $-2^{-1/2}$. See the left side of Figure 1.12.

Figure 1.12. Shown to the left is a depiction for the conclusion of 3.14.
Shown to the right is a depiction for 3.16 (where $\mathsf{T}$ and
$\mathsf{R}$ are not drawn to scale).

######  Theorem 3.14.

Adopting 3.13, there exist constants $c=c(B)>$ and $C=C(B)>1$ such that the
following holds with probability at least $1-Cn^{-10}$ whenever $\delta<c$. If

(3.11) $\displaystyle
G(t,x)=\mathfrak{G}(t,x+1)=-2^{-1/2}t^{2}-2^{-7/6}(3\pi)^{2/3}(x+1)^{2/3},\qquad\text{for
each $(t,x)\in[-\xi,\xi]\times[0,1]$},$

then for each integer $j\in\llbracket n/3,2n/3\rrbracket$, there exists a
(random) twice-differentiable function
$h_{j}:[-\xi/2,\xi/2]\rightarrow\mathbb{R}$ with

$\displaystyle\displaystyle\sup_{|s|\leq\xi/2}\big{|}\partial_{s}^{2}h_{j}(s)+2^{-1/2}\big{|}\leq\delta^{1/8}+(\log
n)^{-1/4},\qquad\text{and}\qquad\|h_{j}\|_{\mathcal{C}^{1}}\leq C,$

such that

$\displaystyle\displaystyle\sup_{|s|\leq\mathsf{T}/2}\big{|}\mathsf{x}_{j}(s)-n^{2/3}\cdot
h_{j}(n^{-1/3}s)\big{|}\leq n^{-1/5}.$

####  3.4. Brownian Bridges Above a Curve

We will frequently make use of the process obtained by examining the Airy line
ensemble $\bm{\mathcal{A}}$ at a given time, called the Airy point process.

######  Definition 3.15.

Let $\bm{\mathcal{A}}$ denote the Airy line ensemble, as in Definition 2.4.
The random infinite sequence
$\bm{\mathfrak{a}}=(\mathfrak{a}_{1},\mathfrak{a}_{2},\ldots)=\big{(}\mathcal{A}_{1}(0),\mathcal{A}_{2}(0),\ldots\big{)}$
of decreasing real numbers is the _Airy point process_.

In this section we state two results. The first indicates that, under certain
conditions, the gaps between paths at a single time for an ensemble of non-
intersecting Brownian bridges converge to those of the Airy point process. We
begin by describing these conditions more precisely through the following
assumptions. The first imposes that the second derivative of its lower
boundary $f$ is nearly constant; the second additionally imposes that the
entrance data for the ensemble satisfies a global law (analogous to Definition
3.9).

######  Assumption 3.16.

Let $n\geq 1$ be an integer;
$\bm{\delta}=(\delta_{1},\delta_{2},\ldots)\subset(0,1/4)$ be a non-increasing
sequence of real numbers satisfying $\lim_{k\rightarrow\infty}\delta_{k}=0$
and $\delta_{k}\geq k^{-1/10}$ for each integer $k\geq 1$; and
$\mathsf{T}=\mathsf{T}_{n}$ be a real number such that
$\delta_{n}^{-1}n^{1/3}\leq\mathsf{T}\leq n^{1/2}$. Further set
$\mathsf{R}=\mathsf{R}_{n}=n^{20}\geq\mathsf{T}$, and fix a function
$f=f_{n}:[-\mathsf{T},\mathsf{R}]\rightarrow\mathbb{R}$ such that there exists
a function $h=h_{n}:[-\mathsf{T},\mathsf{T}]\rightarrow\mathbb{R}$ satisfying

(3.12)
$\displaystyle\sup_{s\in[-\mathsf{T},\mathsf{T}]}\big{|}\partial_{s}^{2}h(s)+2^{-1/2}\big{|}\leq\delta_{n};\qquad\displaystyle\sup_{s\in[-\mathsf{T},\mathsf{T}]}\big{|}f(s)-h(s)\big{|}\leq\delta_{n}.$

Let $\bm{u}=\bm{u}^{n}\in\mathbb{W}_{n}$ and
$\bm{v}=\bm{v}^{n}\in\mathbb{W}_{n}$ be sequences such that

$\displaystyle f(-\mathsf{T})\leq u_{n}\leq u_{1}\leq f(-\mathsf{T})+n;\qquad
f(\mathsf{R})\leq v_{n}\leq v_{1}\leq f(\mathsf{R})+n.$

Sample non-intersecting Brownian bridges
$\bm{\mathsf{x}}=\bm{\mathsf{x}}^{n}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T},\mathsf{R}]\big{)}$ under
$\mathsf{Q}_{f}^{\bm{u};\bm{v}}$. See the right side of Figure 1.12.

######  Assumption 3.17.

Adopt 3.16. For any $t\in[-\mathsf{T},\mathsf{R}]$, define the event

(3.13)
$\displaystyle\mathscr{F}(t)=\mathscr{F}_{n}(t)=\bigg{\\{}\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}\mathsf{x}_{j}(t)-2^{-7/6}(3\pi)^{2/3}(n^{2/3}-j^{2})-f_{n}(t)\big{|}\leq\delta_{n}n^{2/3}\bigg{\\}}.$

Then, we have $\mathbb{P}\big{[}\mathscr{F}_{n}(t)\big{]}\geq 1-\delta_{n}$
for each real number $t\in[-n^{1/3},n^{1/3}]$.

Similarly to (3.10), 3.17 imposes that we knew in advance that the curves in
$\bm{\mathsf{x}}^{n}$ likely approximate a specific deterministic function at
intermediate times $t$ in the domain. In our eventual context, this knowledge
will come from 3.10.

The following theorem indicates that, under these two assumptions, the gaps
between the bridges in $\bm{\mathsf{x}}^{n}$ converge to those of the Airy
point process $\bm{\mathfrak{a}}$ (from Definition 3.15). It is established in
Section 8.2 below.

######  Theorem 3.18.

Adopt 3.17, and fix an integer $k\geq 1$ and a real number $t\in\mathbb{R}$.
As $n$ tends to $\infty$, the $k$-tuple of gaps

$\displaystyle
2^{1/2}\cdot\big{(}\mathsf{x}_{1}(t)-\mathsf{x}_{2}(t),\mathsf{x}_{2}(t)-\mathsf{x}_{3}(t),\ldots,\mathsf{x}_{k}(t)-\mathsf{x}_{k+1}(t)\big{)},$

converges in law to that
$(\mathfrak{a}_{1}-\mathfrak{a}_{2},\mathfrak{a}_{2}-\mathfrak{a}_{3},\ldots,\mathfrak{a}_{k}-\mathfrak{a}_{k+1})$
of the Airy point process $\bm{\mathfrak{a}}$.

The second result of this section indicates that, if the infinite ensemble
$\bm{\mathcal{L}}$ from 2.8 has the property that the gaps between its paths
at any time converges to those of the Airy point process, then
$\bm{\mathcal{L}}$ must be a parabolic Airy line ensemble, up to a (possibly
random) affine shift. It is established in Section 9.1 below.

######  Proposition 3.19.

Adopt 2.8. Further assume for any integer $k\geq 1$ and real number
$t\in\mathbb{R}$ that the $k$-tuple of gaps
$2^{1/2}\cdot\big{(}\mathcal{L}_{1}(t)-\mathcal{L}_{2}(t),\mathcal{L}_{2}(t)-\mathcal{L}_{3}(t),\ldots,\mathcal{L}_{k}(t)-\mathcal{L}_{k+1}(t)\big{)}$
has the same law as that
$(\mathfrak{a}_{1}-\mathfrak{a}_{2},\mathfrak{a}_{2}-\mathfrak{a}_{3},\ldots,\mathfrak{a}_{k}-\mathfrak{a}_{k+1})$
of the Airy point process $\bm{\mathfrak{a}}$. Then there exist two random
variables $\mathfrak{l},\mathfrak{c}\in\mathbb{R}$, and a parabolic Airy line
ensemble
$\bm{\mathcal{R}}=(\mathcal{R}_{1},\mathcal{R}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ independent from them, such that
$\mathcal{L}_{j}(t)=2^{-1/2}\cdot\mathcal{R}_{j}(t)+\mathfrak{l}t+\mathfrak{c}$
for each $(j,t)\in\mathbb{Z}_{\geq 1}\times\mathbb{R}$.

Let us mention that, although 3.19 as stated is a result about the line
ensemble $\bm{\mathcal{L}}$ on the infinite line $\mathbb{R}$, it will quickly
be reduced to one about line ensembles on finite intervals (see 9.1 below),
which is our reason for including it here.

####  3.5. Proof of 2.9