# Basic properties of the Airy line ensemble

Duncan Dauvergne and Bálint Virág

##  1 Introduction

The Airy line ensemble is a central object in random matrix theory, last
passage percolation, and more generally, for problems about the Kardar-Parisi-
Zhang universality class. It was first constructed by Prähofer and Spohn [15]
as the limit of the polynuclear growth model (PNG), and later as the limit of
nonintersecting Brownian motions, see [2], [3].

The Airy line ensemble $\mathcal{A}$ is a decreasing sequence of
nonintersecting continuous functions
$\mathcal{A}_{1}>\mathcal{A}_{2}>\mathcal{A}_{3}\dots$ defined so that the
stationary Airy line ensemble
$\mathcal{R}_{\cdot}(t)=\mathcal{A}_{\cdot}(t)+t^{2}$ is a determinantal
process with kernel (2).111Note that some authors refer to $\mathcal{A}$ as
the parabolic Airy line ensemble and $\mathcal{R}$ simply as the Airy line
ensemble. For any fixed $t$, the distribution of $\mathcal{R}_{1}(t)$ is
Tracy-Widom, and

$\mathcal{A}_{k}(0)\sim-(3\pi k/2)^{2/3}\qquad\text{ as }k\to\infty.$

The determinantal formula (2) for $\mathcal{A}$ is useful for the definition,
to get convergence, and to prove some properties of fixed-time distributions.
However, it is difficult to deduce even the most basic path properties, such
as continuity, from it directly, see [15].

A useful technique, called the Brownian Gibbs property, was developed by
Corwin and Hammond [3]. This property says that inside any region,
conditionally on the outside of the region, the Airy line ensemble is just a
sequence of independent Brownian bridges of variance $2$ conditioned so that
everything remains nonintersecting and continuous.222In a previous version of
this paper, we had chosen the scaling of $\mathcal{A}$ so that the Brownian
bridges had variance $1$ rather than $2$. The choice of variance $2$ is to
align with convention.

This means that if the boundary is well understood, then one can use the
Brownian bridges to deduce path properties. This is a big if: for a
rectangular region, left and right boundary points can be “jammed” close
together, making non-intersection difficult. Also, the bottom and/or top
boundaries are paths, whose properties are not easily accessible from the
determinantal structure.

The goal of this paper is to tackle these issues and make the Airy line
ensemble more workable for probabilists. Among other things, we obtain a
structural theorem that quantitatively relates the Airy lines to independent
Brownian bridges, modulus of continuity bounds for Airy lines that are optimal
up to a logarithmic factor in the number of lines, tight control of the line
locations, and exponential moment bounds for the number of jammed points on
the boundary. The construction of the scaling limit of Brownian last passage
percolation [6] relies heavily on these results.

The Brownian Gibbs property suggests that one could construct the top lines of
the Airy line ensemble by sampling points on a fine space-time grid according
to the determinantal formula, then connecting them with independent Brownian
bridges that will not intersect because of the fineness of the grid. Indeed,
we have such a result, with one difference: when a group of endpoints are
close together, we have to condition the Brownian bridges between those
endpoints not to intersect. However, we have good control over the size of
these groups of close endpoints. In particular, they will remain bounded as we
include more and more lines in the scale that we are working with. The close
endpoint phenomenon is not a deficiency in our method; close endpoints really
do exist in the Airy line ensemble at the scales we work with.

To make this more precise, pick parameters $\ell,k\in\mathbb{N}$ and
$t,\delta>0$. Let $s_{j}=tj/\ell$, and sample the Airy line ensemble at grid
points $\mathcal{A}_{i}(s_{j})$ for $i\in\\{1\dots 2k\\}$ and
$j\in\\{0\dots\ell\\}$ using the determinantal formula. The bridges connecting
these points will be indexed by $i\in\\{1\dots 2k\\}$ and
$j\in\\{1\dots\ell\\}$.

Let $G$ be the random graph on this index set that connects $(i,j)$ to
$(i+1,j)$ if the two points in one of the two prospective endpoint pairs are
within $\delta$ of each other. Now connect up the points with independent,
variance $2$, Brownian bridges $B_{i,j}:[s_{j-1},s_{j}]\to\mathbb{R}$
conditioned not to intersect if the indices are in the same component of $G$.
This yields a new line ensemble $\mathcal{B}$, see Figure 1 for an
illustration.

######  Theorem 1.1.

The total variation distance of $\mathcal{B}$ and $\mathcal{A}$, both
restricted to the top $k$ lines and the interval $[0,t]$, is stretched
exponentially small in $k$ in the right parameter region.

The precise result is as follows. We use a parameter $\gamma>0$ so that the
time between grid points is at most $k^{-2/3-\gamma}$. Since the distance
between the $k$th and $(k+1)$st Airy points is $O(k^{-1/3})$, the $k$th line
in $\mathcal{A}$ only behaves like a Brownian motion up to the time scale
$k^{-2/3}$. At this time it will start interacting with nearby lines, so we
always need our grids to have spacing $o(k^{-2/3})$ to have any hope of
proving a result.

We show that for $\gamma\in[c_{2}\log(\log k)/\log k,2]$, as long as $k\geq
3$, $t>0$, $\delta=k^{-1/3-\gamma/4}$ and $\ell\geq tk^{2/3+\gamma}$, that the
total variation distance between $\mathcal{B}$ and $\mathcal{A}$ restricted to
the top $k$ lines and the interval $[0,t]$ is bounded above as follows:

$d_{TV}(\mathcal{B}|_{\\{1,\dots,k\\}\times[0,t]},\mathcal{A}|_{\\{1,\dots,k\\}\times[0,t]})\leq\ell\exp(-c_{1}\gamma
k^{\gamma/12}).$

Here $c_{1}$ and $c_{2}$ are universal constants. Note that we don’t need to
consider a lower boundary, which is one of the main difficulties when applying
the Brownian Gibbs property. The reason for this is that we have taken into
account point locations from $1,\dots,2k$, and so the point locations for the
lines $k+1,\dots,2k$ play the role of the lower boundary. The value $2k$ is
chosen just for convenience; a smaller value could be used if desirable.

(a)

(b)

Figure 1: An illustration of the bridge representation $\mathcal{B}$ of the
Airy line ensemble. Figure 1(a) is the Airy line ensemble, with points at
three times identified. Points with the same time coordinate are grouped
together if they are close. To sample the bridge representation on this grid,
we erase all lines between the specified points and resample independent
Brownian bridges that are conditioned not to intersect each other if either of
their endpoints are close (i.e. have the same colour). The result is Figure
1(b). Because we ignored the lower boundary condition, there is a reasonable
chance that the lowest bridges will intersect this boundary. However, if we
only look at the top half of the lines in the bridge representation, then with
high probability they do not intersect each other and resemble what we would
have obtained from the usual Brownian Gibbs resampling.

We have good control over the behavior of the graph $G$. In particular, the
probability that a typical vertex is not isolated is $O(k^{-3\gamma/4})$ and
edges in $G$ are typically well spaced (see Proposition 7.4 for a precise
statement). Also, the maximal component size $M_{k}$ of the graph $G$ behaves
well; our general bound in the most common case of fixed $\gamma$ gives the
following.

######  Proposition 1.2.

Fix $\gamma>0$, $t>0$ and let $\ell=\ell_{k}\sim tk^{2/3+\gamma}$ and
$k\to\infty$. Then

$\mathbb{P}(M_{k}\geq 14(1+1/\gamma))\to 0.$

One of the main technical difficulties in proving Theorem 1.1 is understanding
the distribution of pairs of points that are within $\delta$ of each other in
the Airy point process $\\{\mathcal{A}_{k}(0):k\in\mathbb{N}\\}$. In an
interval of the form $[-a,-a+\ell]$, $a\geq 1$ the number $L$ of such pairs is
typically of the order $\eta=a^{2}\delta^{3}\ell$. By analyzing the
determinantal structure of the Airy point process we show that the probability
of being larger than this typical value decays exponentially.

######  Proposition 1.3.

There exists a universal constant $c>0$ such that

$\mathbb{P}(L>m\eta)\leq 2e^{-cm\left(\frac{\eta}{\eta+1}\right)}\qquad\text{
for all }m>0.$

Similar results hold for pairs of close points in all determinantal processes
with nice enough kernels, see Proposition 5.4.

One consequence of the bridge representation is a modulus of continuity bound
for the Airy line ensemble that is optimal up to the power of $\log k$.

######  Theorem 1.4.

There is a constant $d>0$ so that for all $t>0$ there is a random $C>0$ so
that a.s.

$\frac{|\mathcal{A}_{k}(s+r)-\mathcal{A}_{k}(s)|}{\sqrt{r\log(1+1/r)}\log^{d}k}<C$

for all $k\in\mathbb{N}$, $r>0$, $s,s+r\in[0,t]$.

Prähofer and Spohn [15] proved the continuity of the Airy line ensemble.
Hammond [8] also has modulus of continuity bounds that get exponentially worse
with $k$.

On the way to proving these results, we establish several properties of
Brownian bridges conditioned not to intersect. A modulus of continuity bound
can be directly deduced from the following (see Proposition 3.4).

######  Proposition 1.5.

Let $B_{i}$, $i\in\\{1,\ldots k\\}$ be independent Brownian bridges with slope
$b_{i}$ on some time interval $[a,b]$ conditioned not to intersect. Then for
any $i$, $t>0$, $s,s+t\in[a,b]$ we have

$\mathbb{P}(|B_{i}(s+t)-B_{i}(s)-b_{i}t|\geq m\sqrt{kt})\leq ce^{-dm^{2}}.$

Here $c,d>0$ are universal constants.

This bound gets worse when the number of bridges tends to infinity, but the
$\sqrt{k}$ factor is necessary as this is the typical drift when all the
bridges start and end together. We also have a modulus of continuity bound for
the top lines in a Dyson’s Brownian motion
$\\{W^{n}_{1}>W^{n}_{2}>\dots>W^{n}_{n}\\}$ (i.e. a collection of $n$
nonintersecting Brownian motions all started at $0$). Significantly, our bound
does not get worse with $n$ at the scale where Dyson’s Brownian motion
converges to the Airy line ensemble.

######  Proposition 1.6.

Fix $k\in\mathbb{N},c>0$. For every $n\in\mathbb{N}$,
$t>0,\;s\in[0,ctn^{-1/3}]$, and $m>0$ we have

$\mathbb{P}\bigg{(}\left|W_{k}^{n}(t+s)-W_{k}^{n}(t)-s\sqrt{n/t}\right|>m\sqrt{s}\bigg{)}\leq
c_{k}e^{-c_{k}^{\prime}m^{3/2}}.$

This bound, together with the Kolmogorov-Centsov criterion, gives a short
proof of the tightness for the rescaling of Dyson’s Brownian motion
$\mathcal{C}^{n}$ defined in (3) that converges to the Airy line ensemble (the
main result of [3]).

######  Corollary 1.7 ([3]).

For every $k$, $\\{\mathcal{C}^{n}_{k},n\geq 1\\}$ is tight with respect to
the uniform-on-compact topology.

### Related work

The probabilistic study of the Airy line ensemble was initiated by Corwin and
Hammond [3]. The main tools that we use in this paper come from that work,
namely, the Brownian Gibbs property and a monotonicity lemma for
nonintersecting Brownian bridges with respect to their endpoints, see Lemma
2.5.

Hammond [8] used the Brownian Gibbs property to prove Radon-Nikodym derivative
and other regularity bounds for parts of the Airy line ensemble with respect
to Brownian bridges. Subsequent papers of Hammond [10, 9, 11] applied this
work to understanding problems about the geometry of last passage paths in
Brownian last passage percolation and the roughness of limiting growth
profiles in that model. Ergodicity of the Airy line ensemble for time-shifts
was proven by Corwin and Sun [5].

One goal of much of the above work is to characterize the Airy line ensemble
without relying as much on determinantal formulas. Such a characterization
could allow a method for proving convergence results for models within the KPZ
universality class that either have no exact formulas, or only have
intractable formulas. We believe that our work can be used as a starting point
for trying to prove such a characterization.

In particular, Corwin and Hammond [3] conjectured that, up to a value shift,
the Airy line ensemble is the unique nonintersecting line ensemble satisfying
both the Brownian Gibbs property and stationarity after the addition of a
parabola. Even a partial resolution of their conjecture could be used for
proving convergence results. For example, in [4], they outlined how this
conjecture could be used to prove convergence of the KPZ equation to the Airy
process.

### Organization of the paper

The paper is organized as follows. In Section 2, we define the most important
terms. In Section 3 we prove a concentration result about the location of
Dyson lines, and we prove increment tail bounds and a modulus of continuity
result for nonintersecting Brownian bridges. These lemmas are used in later
sections. Section 4 gives tail bounds for increments of the top Dyson’s
Brownian motion lines, proving Proposition 1.6. In Section 5 we study the Airy
point process, the fixed-time marginal of the Airy line ensemble. We recall
and prove theorems about point locations and prove new results about close
points, including Propositions 1.2 and 1.3. In Section 7 we prove the bridge
representation, Theorem 1.1. Section 8 contains the proof of the modulus of
continuity, Theorem 1.4.

##  2 Preliminaries

In this section we recall definitions related to Dyson’s Brownian motion and
the Airy line ensemble.

First, a $k$-level Dyson’s Brownian motion is the limit

$W^{k}=(W^{k}_{1},\dots,W^{k}_{k}):[0,\infty)\to\mathds{R}^{k}$

of $k$ independent standard Brownian motions $W^{k}_{1},\dots,W^{k}_{k}$
conditioned so that

$W^{k}_{1}(s)>W^{k}_{2}(s)>\dots>W_{k}(s)$

for all times $s\in(\epsilon,1/\epsilon)$ as $\epsilon\to 0$. A $k$-level
Dyson’s Brownian motion has the same distribution as the eigenvalues of a
matrix-valued Brownian motion in the space of $k\times k$ Hermitian matrices
with complex entries.

A Brownian $k$-melon

$B^{k}=(B^{k}_{1},\dots,B^{k}_{k}):[0,t]\to\mathds{R}^{k}$

is a system of $k$ independent Brownian bridges $B_{1},\dots,B_{k}$ with
$B^{k}_{i}(0)=B^{k}_{i}(t)$ for all $i\in\\{1,\dots,k\\}$, conditioned in a
similar limiting fashion so that

$B^{k}_{1}(s)>B^{k}_{2}(s)>\dots>B^{k}_{k}(s)$

for all $s\in(0,t)$. We note that analogously to the usual relationship
between Brownian bridge and Brownian motion, we have that

$B^{k}(s)\stackrel{{\scriptstyle
d}}{{=}}\frac{t-s}{\sqrt{t}}W^{k}\left(\frac{s}{t-s}\right),$ (1)

where the equality above holds in distribution in the space of $k$-tuples of
random functions.

We say that Brownian motion (or $k$-melon, bridge, or Dyson’s Brownian motion)
has variance $v$ if its quadratic variation in an interval $[s,t]$ is
proportional to $v(t-s)$.

The top lines of an $n$-level Dyson’s Brownian motion (or alternatively, a
Brownian $n$-melon) converge in law to a limit called the Airy line ensemble
as $n\to\infty$. To state this convergence precisely, we first discuss line
ensembles and define the limiting process.

### Line ensembles

The Airy line ensemble is a determinantal process, first introduced by
Prähofer and Spohn [15]. The existence of a continuous version of this process
was proven by Corwin and Hammond [3] using the Brownian Gibbs property of
Dyson’s Brownian motion and the Airy line ensemble.

Let $\mathcal{S}$ be the space of (possibly finite) sequences of continuous
functions $f=(f_{1},f_{2},\dots)$, where each $f_{i}:I\to\mathds{R}$ for some
closed interval $I\subset\mathds{R}$. The domain $I$ and the number of
functions may depend on the element of $\mathcal{S}$.

We say that a sequence $f^{n}=(f^{n}_{1},f^{n}_{2},\dots)\in\mathcal{S}$
converges to a limiting function $f=(f_{1},\dots)$ if $f^{n}_{i}\to f_{i}$
uniformly on compact subsets of $\mathds{R}$ for all $i$. For this definition,
we do not require that $f^{n}$ and $f$ have the same number of functions or
that the functions have the same domains. We only require that $f_{k}$ is
defined if and only if $f^{n}_{k}$ is defined for all large enough $n$, and
that for any $a$ in the domain of $f^{k}$, that $a$ is in the domain of
$f^{n}_{k}$ for all large enough $n$.

######  Definition 2.1.

A line ensemble $\mathcal{L}=(\mathcal{L}_{1},\mathcal{L}_{2},\dots)$ is a
random element of $\mathcal{S}$. We say that a line ensemble is ordered if
almost surely,

$\mathcal{L}_{i}(x)\geq\mathcal{L}_{i+1}(x)\qquad\text{ for all
}i\in\mathbb{N},x\in\mathds{R}.$

We say that $\mathcal{L}$ is strictly ordered if strict inequality can replace
weak inequality above for all $i,x$.

We write $\mathcal{L}|_{\\{i,\dots,k\\}\times[c,d]}$ for the sequence
$(\mathcal{L}_{i},\dots,\mathcal{L}_{k})$ restricted to the interval $[c,d]$.

######  Definition 2.2.

An ordered line ensemble $\mathcal{L}$ satisfies the Brownian Gibbs property
(with variance $v$) if the following holds for all $k,\ell\in\mathbb{N}$ and
$c<d\in\mathds{R}$. Let $\mathcal{F}$ be the $\sigma$-algebra generated by the
set

$\\{\mathcal{L}_{i}(x):(i,x)\notin\\{k+1,\dots,k+\ell\\}\times[c,d]\\}.$

Then the conditional distribution of
$\mathcal{L}|_{\\{k+1,\dots,k+\ell\\}\times[c,d]}$ given $\mathcal{F}$ is
equal to the law of $\ell$ independent, variance $v$, Brownian bridges
$B_{1},\dots,B_{\ell}:[c,d]\to\mathds{R}$ with $B_{i}(c)=\mathcal{L}_{k+i}(c)$
and $B_{k+i}(d)=\mathcal{L}_{i}(d)$ for all $i\in\\{1,\dots,\ell\\}$,
conditioned on the event

$\mathcal{L}_{k}(r)>B_{1}(r)>B_{2}(r)>\dots>B_{\ell}(r)>\mathcal{L}_{k+\ell+1}(r)\qquad\text{
for all }r\in[c,d].$

If $\mathcal{L}_{k}$ or $\mathcal{L}_{k+\ell+1}$ don’t exist, then drop the
corresponding inequality from the conditioning.

Rather than repeating the above statements throughout the paper to describe a
sequence of Brownian bridges with the above properties, we will simply say
that $B_{1},\dots,B_{\ell}$ is a sequence of Brownian bridges with endpoints
$B_{i}(c)=\mathcal{L}_{k+i}(c)$ and $B_{i}(d)=\mathcal{L}_{k+i}(d)$
conditioned to avoid each other and the boundaries
$\mathcal{L}_{k},\mathcal{L}_{k+\ell+1}$.

Note that both Dyson’s Brownian motion and Brownian $k$-melons have the
Brownian Gibbs property, and that the Brownian Gibbs property is preserved by
taking limits in the space of ordered line ensembles.

######  Definition 2.3.

The Airy line ensemble $\mathcal{A}$ is a continuous line ensemble with lines
$\mathcal{A}_{i}$ indexed by $\mathbb{N}$ defined by the requirement that the
stationary Airy line ensemble $\mathcal{R}(t)=\mathcal{A}(t)+t^{2}$ is a
determinantal process with kernel

$K((x,s);(y,t))=\begin{cases}\int_{0}^{\infty}d\lambda
e^{-\lambda(s-t)}\operatorname{Ai}(x+\lambda)\operatorname{Ai}(y+\lambda),\qquad&\;\text{if}\;s\geq
t,\\\ -\int_{-\infty}^{0}d\lambda
e^{-\lambda(s-t)}\operatorname{Ai}(x+\lambda)\operatorname{Ai}(y+\lambda),\qquad&\;\text{if}\;s<t,\end{cases}$
(2)

where $\operatorname{Ai}(\cdot)$ is the Airy function.

Prahöfer and Spohn [15] first constructed the Airy line ensemble as the limit
of the polynuclear growth model. Adler and van Moerbeke [2] then showed that
the finite dimensional distributions of a rescaled $n$-level Dyson’s Brownian
motion converges to those of the Airy line ensemble. Corwin and Hammond [3]
showed that this convergence takes place in the space of ordered line
ensembles. They also showed that the Airy line ensemble is unique, and that a
shifted version satisfies the Brownian Gibbs property. To describe these
theorems, we first introduce the scaling of Dyson’s Brownian motion.

Let $(W^{n}=(W^{n}_{1},\dots,W^{n}_{n}))_{n\in\mathds{Z}}$ be a sequence of
$n$-level Dyson’s Brownian motions. For $k<n$, define

$\mathcal{C}^{n}_{k}(t)=\left(W^{n}_{k}(1+2tn^{-1/3})-2\sqrt{n}-2tn^{1/6}\right)n^{1/6}.$
(3)

Let $\mathcal{C}^{n}$ be the line ensemble with $n$ lines whose $k$th line is
given by $\mathcal{C}^{n}_{k}$.

######  Theorem 2.4 ([15], [2], [3]).

The Airy line ensemble $\mathcal{A}$ is the distributional limit of the line
ensembles $\mathcal{C}^{n}$ as $n\to\infty$. Moreover, $\mathcal{A}$ has the
Brownian Gibbs property with variance $2$ and is a strictly ordered line
ensemble with probability $1$.

Figure 2: Dyson’s Brownian motion from time $0$ to time $1$. If we zoom in
around the location $(1,2\sqrt{n})$ on a $O(n^{-1/3})\times O(n^{-1/6})$ box,
then we get the Airy line ensemble.

Note that the line ensemble $\mathcal{C}^{n}$ has the Brownian Gibbs property
since it is an affine shift of a Dyson’s Brownian motion. This immediately
implies that $\mathcal{A}$ has the Brownian Gibbs property. As suggested by
the name, the stationary Airy line ensemble $\mathcal{R}$ is stationary in
time, as can be seen from the kernel formula (2). We also note that
$\mathcal{R}$ has the same distribution as its time reversal
$\mathcal{R}(-\;\cdot)$.

The main obstacle in the paper [3] for proving Theorem 2.4 was in showing
tightness of the line ensembles $\mathcal{C}^{n}$. We give a simple proof of
this fact in the last section of the paper, based on a proof of a modulus of
continuity for lines in the stationary Airy line ensemble.

We also record here an intuitive lemma from [3] which gives monotonicity in
the endpoints and boundary conditions for non-intersecting Brownian bridge
ensembles. For this lemma, let $\mathds{R}^{k}_{\geq}$ be the set of
$\mathbf{x}\in\mathds{R}^{k}$ such that $x_{i}\geq x_{i+1}$ for all
$i\in\\{1,\dots,k-1\\}$.

######  Lemma 2.5 (Lemmas 2.6 and 2.7, [3]).

Fix $k\in\mathbb{N}$ and $a<b\in\mathds{R}$. Let
$\mathbf{w},\mathbf{x},\mathbf{y},\mathbf{z}\in\mathds{R}^{k}_{>}$ be such
that $w_{i}\geq x_{i}$ and $y_{i}\geq z_{i}$ for all $i$. Also, let
$f,g:[a,b]\to\mathds{R}\cup\\{-\infty\\}$ be such that

$f(a)\leq x_{k},\;\;g(a)\leq w_{k}\quad\;\text{and}\;\quad f(b)\leq
z_{k},\;\;g(b)\leq y_{k}.$

Finally, assume that $f(r)\leq g(r)$ for all $r\in[a,b]$. Then there exists a
$2k$-tuple of random functions $(B_{1},\dots,B_{k},C_{1},\dots,C_{k})$ where
each function has domain $[a,b]$ such that the following holds:

  1. (i)

The sequence $(B_{1},\dots,B_{k})$ has the distribution of $k$ Brownian
bridges with $B_{i}(a)=x_{i}$ and $B_{i}(b)=z_{i}$, conditioned to avoid $f$
and each other.

  2. (ii)

The sequence $(C_{1},\dots,C_{k})$ has the distribution of $k$ Brownian
bridges with $C_{i}(a)=w_{i}$ and $C_{i}(b)=y_{i}$, conditioned to avoid $g$
and each other.

  3. (iii)

$C_{i}(r)\geq B_{i}(r)$ for all $r\in[a,b]$.

In the above lemma, the definition of nonintersecting bridges starting or
ending at the same location should be understood in the same limiting sense as
in the definition of Dyson’s Brownian motion. Note also that we can consider
the case of no lower boundary condition on the bridges (that is, $f$ or $g$
equal to $-\infty$). We will also use the limiting case when the endpoints of
a few top or bottom bridges are taken to $\pm\infty$, essentially removing
them from the conditioning.

##  3 Nonintersecting Brownian ensembles

We first prove a bound on the deviation of the $k$th point in Dyson’s Brownian
motion. Recall that the points at time $1$ in a Dyson’s Brownian motion are
equal in distribution to the eigenvalues of the Gaussian unitary ensemble.
Ledoux and Rider [14] show the $k=1$ case of the following theorem. Their
proof extends to general $k$. These results hold for the $\beta\geq 1$ Hermite
random matrix ensembles. Note that this theorem can also be deduced from
bounds coming from determinantal formulas (i.e. see [7]).

######  Theorem 3.1.

Let $W^{n}_{k}$ be the $k$th line of an $n$-level Dyson’s Brownian motion.
There exist constants $c_{k}$ and $d_{k}$ such that for all $m>0$ and all
$n\geq 1$ we have

$\mathbb{P}(|W^{n}_{k}(1)-2\sqrt{n}|\geq mn^{-1/6})\leq
c_{k}e^{-d_{k}\max(m^{3/2},n^{-1/3}m^{2})}.$

###### Proof.

The $k=1$ case is proved in [14], see equation (2.7) there. It implies the
upper tail bound on $W^{n}_{k}(1)-2\sqrt{n}$ for the general $k$ case by
monotonicity. It also implies the lower tail bound for when $m\geq 8n^{2/3}$,
since $W^{n}_{n}\stackrel{{\scriptstyle d}}{{=}}-W^{n}_{1}$, and we can extend
this to all $m\geq(k+1)^{-1}n^{2/3}$ by changing the constants $c_{k}$ and
$d_{k}$.

For the lower tail bound, Ledoux and Rider [14] (p. 1331) first construct an
easily tractable tridiagonal matrix $H$ whose eigenvalues are dominated by
$W^{n}_{k}(1),$ $k=1,\ldots,n$. Second, they construct a deterministic test
vector $v$ so that the event

$A=\\{\langle v,Hv\rangle/\langle v,v\rangle>2\sqrt{n}-mn^{-1/6}\\}$

has $\mathbb{P}A\geq 1-ce^{-dm^{3/2}}$. Their test vector is supported on the
first $mn^{1/3}$ coordinates. When $mn^{-1/3}\leq(k+1)^{-1}n$, their argument
can be extended to construct test vectors $v_{1}\ldots v_{k}$ with supports at
least distance $2$ from each other, so that the corresponding events $A_{i}$
have probabilities similarly bounded below with different constants. Shifting
down the non-zero coordinates of original vector $v$ works.

Consider the projection $PHP$ of the matrix $H$ to the space with basis given
by $v_{1},\dots,v_{k}$. Because $H$ is tridiagonal and the supports of the
$v_{i}$ are separated, $PHP$ is diagonal in this basis with elements $\langle
v_{i},Hv_{i}\rangle/\langle v_{i},v_{i}\rangle$. Projections do not increase
$k$th eigenvalues, so we get $W_{k}(1)>2\sqrt{n}-mn^{-1/6}$ on the event
$\bigcap A_{i}$, as required. ∎

Our first use of this theorem is a modulus of continuity for Brownian bridges
conditioned not to intersect.

######  Lemma 3.2.

There exist constants $c,d$ so that the following holds. Let $k\in\mathbb{N}$,
$B_{i}$, $i\in\\{1,\ldots k\\}$ be Brownian bridges with arbitrary start and
end points, and slope $b_{i}$ on some time interval $[a,b]$ conditioned not to
intersect. Then for any $i$, $t>0$, $s,s+t\in[a,b]$ we have

$\mathbb{P}(|B_{i}(s+t)-B_{i}(s)-b_{i}t|\geq m\sqrt{kt})\leq ce^{-dm^{2}}.$

###### Proof.

By Brownian scaling, we may assume that $[a,b]=[0,1]$, $B_{i}(0)=B_{i}(1)=0$
for the $i$ in question, and that the start and endpoints of the $B_{i}$ are
in a decreasing order. Also, by the time and value-reversal symmetry of the
problem, we may assume $s\leq 1/2$ and for the price of a factor of 2 we can
just prove the upper bound on $B_{i}(s+t)-B_{i}(s)$.

For this, condition on the values of all the $B_{j}$ at time $s$. The
remaining process is again Brownian bridges conditioned not to intersect. By
Lemma 2.5, the conditional law of $B_{i}(s+t)$ can be dominated by moving the
starting and ending points (at times $s$ and $1$) of all the other Brownian
bridges up while keeping their order. Those above $B_{i}$ are moved to
$(\infty,\infty)$ and those below to $(B_{i}(s),B_{i}(1))$. So on $[s,1]$ the
process $B_{i}$ can be coupled monotonously with a linear function from
$B_{i}(s)$ to $B_{i}(1)$ plus the top line of an independent $k-i+1$-melon on
the interval $[s,1]$. Using the equivalence (1), there is therefore a coupling
so that

$B_{i}(s+t)-B_{i}(s)\leq\frac{-B_{i}(s)t}{1-s}+\sqrt{\frac{t(1-s-t)}{1-s}}X$

where $X$ has the same distribution as $W^{k-i+1}_{1}(1)$, the top line of the
$k-i+1$ level Dyson’s Brownian motion at time 1 and is independent of
$B_{i}(s)$. The worst case here is $i=1$. We can similarly use (1) to relate
the value of $-B_{i}(s)$ to $-W^{k}_{i}(1)$. The worst case value of $i$ here
is $i=k$, in which case $-W^{k}_{k}(1)\stackrel{{\scriptstyle
d}}{{=}}W^{k}_{1}(1)$. So with $Y$ independent of $X$ and having the same
distribution as $W^{k}_{1}(1)$ we have that in some coupling

$B_{i}(s+t)-B_{i}(s)\leq\frac{t\sqrt{s(1-s)}}{1-s}Y+\sqrt{\frac{t(1-s-t)}{1-s}}X\leq\sqrt{t}(X^{+}+Y^{+}).$

The claim now follows from the bound of Theorem 3.1. ∎

Modulus of continuity results naturally follow from statements such as the one
in Lemma 3.2. The classical example of this is Lévy’s modulus of continuity of
Brownian motion. For future use, we state this next lemma in greater
generality than we need it here.

The proof mimics the proof of Levy’s modulus of continuity of Brownian motion.

######  Lemma 3.3.

Let $T=I_{1}\times\dots\times I_{d}$ be a product of bounded real intervals of
length $b_{1},\dots,b_{d}$. Let $c,a>0$. Let $\mathcal{H}$ be a random
continuous function from $T$ taking values in a vector space $V$ with norm
$|\cdot|$. Assume that for every $i>0$, there exists
$\alpha_{i}\in(0,1),\beta_{i},r_{i}>0$ such that

$\mathbb{P}(|\mathcal{H}(t+e_{i}r)-\mathcal{H}(t)|\geq mr^{{\alpha_{i}}})\leq
ce^{-a{m^{{\beta_{i}}}}}$ (4)

for every coordinate vector $e_{i}$, every $m>0$ with $t,t+re_{i}\in T$ and
$r<r_{i}$. Set $\beta=\min_{i}\beta_{i},\alpha=\max_{i}\alpha_{i}$, and
$b=\max_{i}b$. Then with probability one we have

$|\mathcal{H}(t+s)-\mathcal{H}(t)|\leq
C\left(\sum_{i=1}^{d}|s_{i}|^{\alpha_{i}}\log^{1/\beta_{i}}\left(\frac{2b^{\alpha/\alpha_{i}}}{|s_{i}|}\right)\right),$
(5)

for every $t,t+s\in T$ with $|s_{i}|\leq r_{i}$ for all $i$. Here $C$ is
random constant satisfying

$\mathbb{P}(C>m)\leq cc_{0}e^{-c_{1}m^{\beta}},$

where $c_{0}$ and $c_{1}$ are constants that depend on
$\alpha_{1},\dots\alpha_{d},\beta_{1},\dots,\beta_{d},d$ and $a$. Notably,
they do not depend on $b$ or $c$.

Note that the above lemma can also be extended to the case when
$\alpha_{i}=1$, but the power of $1/\beta_{i}$ in the logarithm term changes
to a power of $1+1/\beta_{i}$.

###### Proof.

We first consider the case when $T=\prod_{i=1}^{d}[0,1]$. We will make the
simplifying assumption that $r_{i}=1$ for all $i$. If this is not the case,
then we can simply start our discretization on finer grid, and the result
follows in the same way.

We will prove the bound when $s$ is a multiple of the unit vector $e_{i}$ for
some $i\in\\{1,\dots,d\\}$. The result then follows by the triangle
inequality. For ease of notation, we set $i=1$. Let $k\in\mathbb{N}$ be large
enough so that $k\alpha_{j}>\alpha_{1}$ for all $j\in\\{2,\dots,d\\}$ Let

$P_{n}^{1}=\\{0,1/2^{n},1/2^{n},3/2^{n},\ldots,1\\}\quad\;\text{and}\;\quad
P_{n}^{j}=\\{0,1/2^{nk},2/2^{nk},\ldots,1\\}$

for $j\in\\{2,\dots,d\\}$, and set $P_{n}=\prod_{i=1}^{d}P_{n}^{i}$. Define
the set of translated boxes

$C_{n}=\left\\{S=v+[0,1/2^{n}]\times\prod_{j=2}^{d}[0,1/2^{nk}]:v\in
P_{n},S\subset T\right\\}.$

For $n\in\\{0,1,\dots\\}$, define “linear” approximations $\mathcal{H}_{n}$ of
$\mathcal{H}$, by setting $\mathcal{H}_{n}=\mathcal{H}$ on $P_{n}$, and by
requiring that $\mathcal{H}_{n}$ is a polynomial with degree at most $1$ in
each variable on each box $D\in C_{n}$. This description uniquely defines
$\mathcal{H}_{n}$ as a continuous function with the property that for any
$D\in C_{n}$, $\mathcal{H}_{n}$ is linear in each coordinate. Moreover,
$\mathcal{H}_{n}\to\mathcal{H}$ uniformly on $T$. We also set
$\mathcal{H}_{-1}=0$.

Now let $t,t+s\in T$ be vectors that differ only on the first coordinate.
Setting $\mathcal{G}_{n}=\mathcal{H}_{n}-\mathcal{H}_{n-1}$, we can write
$\mathcal{H}(t+s)-\mathcal{H}(t)=\sum_{n=0}^{\infty}\mathcal{G}_{n}(t+s)-\mathcal{G}_{n}(t)$.
Letting $||\cdot||$ denote the uniform norm on functions, by linearity of
$\mathcal{H}_{n}$ in the first coordinate on every box $S\in C_{n}$ we have
$\|\partial_{1}\mathcal{G}_{n}\|\leq 2^{n}\|\mathcal{G}_{n}\|$. Therefore
since $s$ only has a non-zero first coordinate, by the the mean value theorem,
we have

$|\mathcal{G}_{n}(t+s)-\mathcal{G}_{n}(t)|\leq|s|2^{n}\|\mathcal{G}_{n}\|.$
(6)

Moreover, the triangle inequality gives that

$|\mathcal{G}_{n}(t+s)-\mathcal{G}_{n}(t)|\leq 2\|\mathcal{G}_{n}\|.$ (7)

Hence to estimate $\mathcal{H}(t+s)-\mathcal{H}(t)$, we need a good bound on
$\|\mathcal{G}_{n}\|$.

For a given box $S\in C_{n-1}$, the values of $\mathcal{H}_{n-1}$ on $S$ are
convex combinations of the values on $\mathcal{H}$ on the vertices of $S$.
Also, the values of $\mathcal{H}_{n}$ on $S$ are convex combinations of the
values of $\mathcal{H}$ on $S\cap P_{n}$. In particular, this implies that

$\max_{t\in S}|\mathcal{H}_{n-1}(t)-\mathcal{H}_{n}(t)|\leq\max_{t+s,t\in
S\cap P_{n}}|\mathcal{H}(t+s)-\mathcal{H}(t)|$

Using this, we get that

$\|\mathcal{G}_{n}\|\leq N_{n,1}+\cdots+N_{n,d},$

where

$\displaystyle N_{n,1}$
$\displaystyle=2\max\\{|\mathcal{H}_{n}-\mathcal{H}_{n-1}|:t+s,t\in
P_{n},|s_{1}|=2^{-n}s_{j}=0,j\geq 2\\}\quad\;\text{and}\;$ $\displaystyle
N_{n,i}$
$\displaystyle=2^{k}\max\\{|\mathcal{H}_{n}-\mathcal{H}_{n-1}|:t+s,t\in
P_{n},|s_{j}|=2^{-nk},s_{i}=0,i\neq j\\}.$

By (4) and a union bound, for all $m>0$ we have that

$\displaystyle\mathbb{P}\left(N_{n,1}>2^{-n\alpha_{1}}m((n+1)kd)^{1/\beta_{1}}\right)$
$\displaystyle\leq ce^{(n+1)kd(\log 2-am^{\beta_{1}})}\quad\;\text{and}\;$ (8)
$\displaystyle\mathbb{P}\left(N_{n,i}>2^{k}2^{-nk\alpha_{i}}2^{-n}m((n+1)kd)^{1/\beta_{i}}\right)$
$\displaystyle\leq ce^{(n+1)kd(\log 2-am^{\beta_{i}})}$ (9)

for $i\in\\{2,\dots,d\\}$. Set

$D=\sup_{n\in\\{0,1,\dots\\}}\frac{||\mathcal{G}_{n}||2^{n\alpha_{1}}}{(n+1)^{1/\beta_{1}}}.$

By the bounds above on $N_{n,j}$ and the fact that $k\alpha_{j}>\alpha_{1}$
for $j\neq 1$, we can conclude that the random variable $D$ is finite almost
surely and satisfies the tail bound

$\mathbb{P}(D\geq m)\leq cc_{0}e^{-c_{1}m^{\beta}}$

where $\beta=\min_{i}\beta_{i}$, and $c_{0}$ and $c_{1}$ are constants that
depends on the terms $\alpha_{i},\beta_{i},a,$ and $d$. Now by using the
bounds in (6) and (7), for every $t+s,t\in T$ which differ only on the first
coordinate, we have

$\displaystyle|\mathcal{H}(t+s)-\mathcal{H}(t)|$
$\displaystyle\leq\sum_{n=0}^{\infty}D(n+1)^{1/\beta_{1}}2^{-n\alpha_{1}}2(|s|2^{n}\wedge
1)$ $\displaystyle\leq
c_{2}D|s|^{\alpha_{1}}\log^{1/\beta_{1}}\left(\frac{2}{|s|}\right)$

for a constant $c_{2}$ that depends on the terms $\alpha_{i},\beta_{i},$ and
$d$. This completes the proof in the case when $T=[0,1]^{d}$. For general
lengths $b_{i}$, note that by translation we can assume that
$T=\prod_{i=1}^{d}[0,b_{i}]$ and by increasing the domain we can assume that
$b_{i}=b$ for all $i$. Let $\alpha=\max_{i}\alpha_{i}$, and define the process

$b^{\alpha}U(t_{1},\dots,t_{d})=\mathcal{H}(b^{\alpha/\alpha_{1}}t_{1},b^{\alpha/\alpha_{2}}t_{2}\dots,b^{\alpha/\alpha_{d}}t_{d}).$

The process $U$ satisfies the same tail bounds as $\mathcal{H}$ and hence
satisfies the conclusion of the lemma. The bound on $U$ on $[0,1]^{d}$ gives
the desired bound on $\mathcal{H}$ on $T$. ∎

Lemmas 3.2 and 3.3 now immediately imply the following.

######  Proposition 3.4.

There exists universal constants $c,d>0$ so that the following holds. Let
$k\in\mathbb{N}$ and let $B_{i}$, $i\in\\{1,\ldots k\\}$ be independent
Brownian bridges with arbitrary start and end points and slope $b_{i}$ on some
time interval $[a,b]$ conditioned not to intersect. Then there are random
constants $C_{i}$ satisfying

$\mathbb{P}(C_{i}>m)\leq ce^{-dm^{2}}$

for any $m>0$, so that for any $i$, $t>0$, $s,s+t\in[a,b]$ we have

$\frac{|B_{i}(s+t)-B_{i}(s)-b_{i}t|}{\sqrt{kt\log(1+t^{-1})}}\leq C_{i}.$

##  4 Modulus of continuity for top Dyson lines

Before proceeding with the main goal of the paper, which is to understand the
Airy line ensemble, we give a proof of tightness of the Dyson lines in the
scaling (3). This follows immediately from the next proposition in conjunction
with the Kolmogorov-Centsov theorem ([13], Corollary 16.9). The ideas in this
proof will be used later in the paper to give a similar modulus of continuity
result for Airy lines.

######  Proposition 4.1.

Fix $k\in\mathbb{N}$ and $c>0$. There exist constants $c_{k},d_{k}>0$ such
that for every $n\in\mathbb{N}$, $t>0,\;s\in[0,ctn^{-1/3}]$, and $m>0$ we have

$\mathbb{P}\bigg{(}\left|W_{k}^{n}(t)-W_{k}^{n}(t+s)-\frac{s\sqrt{n}}{\sqrt{t}}\right|>m\sqrt{s}\bigg{)}\leq
c_{k}e^{-d_{k}m^{3/2}}.$

Throughout the proof, $c_{k}$ and $d_{k}$ will be constants that depend only
on $k$ and $c$, but may change from line to line.

###### Proof.

We can assume that $m>4c^{3/2}$ by possibly changing $c_{k}$ and $d_{k}$. By
Brownian scaling, it suffices to prove the lemma for $t=1$. Now, by time
inversion and Brownian scaling, for any times $p,q\in\mathds{R}$ we have that

$W^{n}_{k}(p)-W^{n}_{k}(q)\stackrel{{\scriptstyle
d}}{{=}}\sqrt{\frac{p}{q}}W^{n}_{k}(q)-\sqrt{\frac{q}{p}}W^{n}_{k}(p).$ (10)

To apply this property, we first define the error $C^{n}_{k}(s)$ by

$W^{n}_{k}(1+s)=\left(2+s-\frac{s^{2}}{4}\right)\sqrt{n}+n^{-1/6}C^{n}_{k}(s).$

Using the property (10) applied to the difference
$W^{n}_{k}(1)-W^{n}_{k}(1+s)$ and Taylor expanding gives that

$W^{n}_{k}(1)-W^{n}_{k}(1+s)+s\sqrt{n}\stackrel{{\scriptstyle
d}}{{=}}W^{n}_{k}(1+s)-s\sqrt{n}-W^{n}_{k}(1)+\phi_{n}(s),$

where $\phi_{n}(s)$ is a random error term that satisfies

$|\phi_{n}(s)|\leq
d\sqrt{n}|s|(s^{2}+n^{-2/3}|C_{k}^{n}(0)|+n^{-2/3}|C_{k}^{n}(s)|)$

for a constant $d$ that depends on the width $c$ of the interval but not on
$n$, or the point $s$. Therefore we can write

$\displaystyle\mathbb{P}\bigg{(}$
$\displaystyle\left|W^{n}_{k}(1)-W^{n}_{k}(1+s)+s\sqrt{n}\right|>m\sqrt{s}\bigg{)}$
$\displaystyle\leq
2\mathbb{P}\left(W^{n}_{k}(1)-W^{n}_{k}(1+s)+s\sqrt{n}>\frac{m}{2}\sqrt{s}\right)+\mathbb{P}\left(\phi_{n}(s)>\frac{m}{2}\sqrt{s}\right).$

By the bound on $\phi_{n}$, the tail bound on $W^{n}_{k}(1)$ established in
Theorem 3.1, and Brownian scaling, we have that

$\displaystyle\mathbb{P}\left(\phi_{n}(s)>\frac{m}{2}\sqrt{s}\right)\leq
c_{k}e^{-d_{k}m^{3/2}}.$

Note that in fact we have much stronger control over the error $\phi_{n}(s)$
when $mn^{-1/3}/\sqrt{s}=O(1)$, but the above bound suffices for the lemma. It
remains to bound

$\mathbb{P}\left(W^{n}_{k}(1)-W^{n}_{k}(1+s)+s\sqrt{n}>\frac{m}{2}\sqrt{s}\right).$
(11)

The method we use here will be applied again in Lemma 6.1. Set

$r=\frac{m}{2\sqrt{sn}},$

and let $L$ be the line with $L(1)=W_{k}^{n}(1)$ and $L(1+r)=W_{k}^{n}(1+r)$.
Note that our assumption that $m>4c^{3/2}$ implies that $r>s$. The Brownian
Gibbs property for Dyson’s Brownian motion and monotonicity (Lemma 2.5)
implies that in the interval $[1,1+r]$, the line $W_{k}^{n}$ stochastically
dominates $L+B$, where $B:[1,1+r]\to\mathds{R}$ is the bottom line of a
Brownian $k$-melon on $[1,1+r]$ (this is what we get after moving the bottom
boundary to $-\infty$ and the top $k-1$ boundary points to $L(1)$ and
$L(1+r)$). Hence (11) is bounded above by

$\displaystyle\mathbb{P}$
$\displaystyle\left([L+B](1)-[L+B](1+s)+s\sqrt{n}>\frac{m\sqrt{s}}{2}\right)$
$\displaystyle\leq\mathbb{P}\left(L(1)-L(1+s)+s\sqrt{n}>\frac{m\sqrt{s}}{4}\right)+\mathbb{P}\left(B(1+s)<-\frac{m\sqrt{s}}{4}\right).$

By Proposition 3.4, the second term is bounded above by $c_{k}e^{-d_{k}m^{2}}$
for constants $c_{k}$ and $d_{k}$. The first term depends only on the points
$W_{k}^{n}(1)$ and $W_{k}^{n}(1+r)$. In particular, we can write it as

$\displaystyle\mathbb{P}$
$\displaystyle\left(W_{k}^{n}(1)-W_{k}^{n}(1+r)+r\sqrt{n}>\frac{mr}{4\sqrt{s}}\right)$
$\displaystyle\leq\mathbb{P}\left(W_{k}^{n}(1)>2\sqrt{n}+\frac{mr}{16\sqrt{s}}\right)+\mathbb{P}\left(W_{k}^{n}(1+r)<(2+r)\sqrt{n}-\frac{\sqrt{n}r^{2}}{4}-\frac{mr}{16\sqrt{s}}\right).$
(12)

In the final line above we have used that $\sqrt{n}r^{2}/4=mr/(8\sqrt{s})$.
Now, since $s$ is bounded by $cn^{-1/3}$, we have that

$\frac{mr}{16\sqrt{s}}\geq\frac{m^{2}}{32c}n^{-1/6}.$

Hence we can bound the two probabilities in (12) using the tail bounds on
Dyson’s Brownian motion established in Theorem 3.1. This gives that (12) is
bounded above by $c_{k}e^{-d_{k}m^{3}}$ for any $s\in[0,cn^{-1/3}]$. This in
turn bounds (11), completing the proof. ∎

We also give an application of this proposition to bound the probability that
the top line of a Dyson’s Brownian motion ever crosses a particular function.
We first note the following consequence of Proposition 4.1 combined with the
modulus of continuity estimate Lemma 3.3.

######  Corollary 4.2.

For every $k$, there exist constants $c_{k}$ and $d_{k}$ such that for all
$t>0,m>0$ and all $n\geq 1$ we have

$\mathbb{P}\bigg{(}\max_{s\in[0,tn^{-1/3}]}\left|W^{n}_{k}(t+s)-W^{n}_{k}(t)-\frac{s\sqrt{n}}{\sqrt{t}}\right|\geq
m\sqrt{s}\bigg{)}\leq c_{k}e^{-d_{k}m^{3/2}}.$

The next proposition can be thought of as a type of ‘law of the iterated
logarithm’ for Dyson’s Brownian. We get a $2/3$-power in the outer logarithm
rather than the power of $1/2$ seen in the usual law of the iterated
logarithm, since the top tail of the top line Dyson’s Brownian has a Tracy-
Widom $3/2$-exponent.

######  Proposition 4.3.

There exist constants $b,c,$ and $d$ such that for all $m>0$ and $n\geq 1$, we
have that

$\mathbb{P}\bigg{(}W^{n}_{1}(t)\leq
2\sqrt{n}\sqrt{t}+\sqrt{t}n^{-1/6}[m+b\log^{2/3}(n^{1/3}\log(t\;\vee\;t^{-1})+1)]\;\;\forall
t\in[0,\infty)\bigg{)}\geq 1-ce^{-dm^{3/2}}.$

Here $a\vee b$ is the maximum of $a$ and $b$.

Throughout the proof, $b,c,$ and $d$ are universal constants that may change
from line to line.

###### Proof.

Set $a_{i}=(1+n^{-1/3})^{i}$ for $i\in\mathds{Z}$. By Theorem 3.1 and Brownian
scaling, for all $n\geq 1$ and $m>0$ we have

$\mathbb{P}\left(W^{n}_{1}(a_{i})\leq
2\sqrt{n}\sqrt{a_{i}}+[m+b\log^{2/3}(|i|+1)]\sqrt{a_{i}}n^{-1/6}\;\;\forall
i\in\mathds{Z}\right)\leq ce^{-dm^{3/2}}.$

By Corollary 4.2, we can extend this bound to all times $t\in[0,\infty)$. Let
$a_{i}(t)$ be the largest $a_{i}$ such that $a_{i}\leq t$ and let $i(t)$ be
the corresponding value of $i$. We get that

$\mathbb{P}\left(W^{n}_{1}(t)\leq
2\sqrt{n}\sqrt{a_{i}(t)}+\frac{(t-a_{i}(t))\sqrt{n}}{\sqrt{a_{i}(t)}}+[m+b\log^{2/3}(|i(t)|+1)]\sqrt{a_{i}(t)}n^{-1/6}\;\;\forall
t\in[0,\infty)\right)$

is bounded above above by $ce^{-dm^{3/2}}.$ Now, by Taylor expansion and the
definition of the sequence $a_{i}$, we have that

$2\sqrt{n}\sqrt{a_{i}(t)}+\frac{(t-a_{i}(t))\sqrt{n}}{\sqrt{a_{i}(t)}}\leq
2\sqrt{n}\sqrt{t}+\frac{n^{-1/6}}{4}.$

Also, $|i(t)|\leq 4n^{1/3}\log(t\vee t^{-1})$ and $\sqrt{a_{i}(t)}\leq
2\sqrt{t}$. Plugging in these bounds above and simplifying the result proves
the proposition. ∎

##  5 Properties of the Airy point process

In this section, we prove a few basic properties about the distribution of the
points

$(\mathcal{A}_{1}(0),\mathcal{A}_{2}(0),\dots).$

This sequence of points is known as the Airy point process. It is
determinantal with locally trace class kernel given by Equation (2) in the
case $s=t$. To simplify notation in the following lemmas, we write
$A_{i}=\mathcal{A}_{i}(0)$, and for $a\in\mathds{R}$, define

$N_{a}=\left|\\{i\in\mathbb{N}:A_{i}\in[-a,\infty)\\}\right|.$

We first recall facts about the expected location of the $k$th point in the
Airy point process and the expectation and variance of $N_{a}$. The facts
about expectation can be easily derived from standard formulas for the Airy
process density (i.e. see formula 1.17 in [16] and discussion thereafter). The
variance bound is more involved, and is proven as Theorem 1 in [16].

For this next lemma and throughout this section, we define

$\kappa=(3\pi/2)^{2/3}.$

######  Lemma 5.1 (Soshnikov [16] p. 494-495).

  1. (i)

$\mathbb{E}A_{k}=-\kappa k^{2/3}+O(1)$ as $k\to\infty$ and
$\mathbb{E}N_{a}=2a^{3/2}/3\pi+O(1)$ as $a\to\infty$.

  2. (ii)

There exists a constant $c>0$ such that for all $a>0$, we have
$\mathbb{P}(N_{-a}>0)\leq c\exp(-\frac{4}{3}|a|^{3/2})$.

  3. (iii)

There exist constants $c_{1}$ and $c_{2}$ such that for all $a\geq 1$, we have
$\operatorname{Var}(N_{a})\leq c_{1}\log a+c_{2}$.

As an immediate consequence of the above lemma, we can bound the fluctuations
of the number of Airy points in an interval.

######  Lemma 5.2.

There exists constants $\alpha,c>0$ such that for every $x\in\mathds{R}$ and
$b>\alpha\log(\max(x,2))$, we have

$\mathbb{P}\left(\left|N_{x}-\frac{2}{3\pi}x_{+}^{3/2}\right|\geq b\right)\leq
ce^{-b}.$

Here $x_{+}=\max(x,0)$.

###### Proof.

By the monotonicity of $N_{x}$ in $x$, it is enough to prove the lemma when
$x\geq 0$. The number of points in any interval in a determinantal point
process with a locally trace class kernel is equal in distribution to a sum of
Bernoulli random variables (see [12], Theorem 4.5.3). Therefore Bernstein’s
inequality gives that

$\mathbb{P}(|N_{x}-\mathbb{E}N_{x}|\geq b)\leq
2\exp\left(\frac{-\frac{1}{2}b^{2}}{\operatorname{Var}(N_{x})+\frac{1}{3}b}\right).$

Applying the bounds on $\mathbb{E}N_{x}$ and $\operatorname{Var}(N_{x})$ from
Lemma 5.1 completes the proof. ∎

We also record a corollary which translates the above lemma into a bound on
the Airy point locations.

######  Corollary 5.3.

There exist $c,\beta>0$ such that for all $i\in\mathds{Z}_{+}$ and
$m>\beta\log(i+1)$, we have that

$\mathbb{P}(|A_{i}+\kappa i^{2/3}|\geq mi^{-1/3})\leq ce^{-m/5}.$

In the proof $c$ is a constant that may change from line to line.

###### Proof.

Fix $m>0$ and $i\in\mathbb{N}$ and let $x_{m}=\kappa i^{2/3}+mi^{-1/3}$. We
have that

$\mathbb{P}(A_{i}<-x_{m})=\mathbb{P}(N_{x_{m}}<i)=\mathbb{P}\left(\frac{2}{3\pi}x_{m}^{3/2}-N_{x_{m}}>\frac{2}{3\pi}x_{m}^{3/2}-i\right).$
(13)

We have the bound

$\frac{2}{3\pi}x_{m}^{3/2}-i>\frac{3}{2\kappa}m>\frac{m}{5}.$

Moreover, letting $\alpha$ be as in Lemma 5.2, there exists a $\beta>0$ such
that the right hand side above bounds $\alpha\log\max(x_{m},2)$ whenever
$m>\beta\log(i+1)$. Applying Lemma 5.2 then shows that

$\mathbb{P}(A_{i}+\kappa i^{2/3}<-mi^{-1/3})\leq ce^{-m/5}.$

Now let $y_{m}=\kappa i^{2/3}-mi^{-1/3}$ and let $y_{m,+}=\max(y_{m},0)$.
Observe that

$\mathbb{P}(A_{i}>-\kappa
i^{2/3}+mi^{-1/3})=\mathbb{P}(N_{y_{m}}>i)=\mathbb{P}\left(N_{y_{m}}-\frac{2}{3\pi}y_{m,+}^{3/2}>i-\frac{2}{3\pi}y_{m,+}^{3/2}\right).$
(14)

We have that

$i-\frac{2}{3\pi}y_{m,+}^{3/2}>\min(i,\kappa^{-1}m)\geq\min(i,m/5).$

Again letting $\alpha$ be as in Lemma 5.2, there exists a $\beta>0$ such that
right hand above bounds $\alpha\log\max(y_{m},2)$ whenever $5i\geq m>\beta\log
i$. Applying Lemma 5.2 then gives

$\mathbb{P}(A_{i}+\kappa i^{2/3}<-mi^{-1/3})\leq ce^{-m/5},$

for $5i\geq m>\beta\log i$. When $m\geq 5i$, by the standard Tracy-Widom tail
bound (Lemma 5.1 (ii)), we have that

$\mathbb{P}(A_{i}>-\kappa i^{2/3}+mi^{-1/3})\leq\mathbb{P}(A_{1}>-\kappa
i^{2/3}+mi^{-1/3})\leq ce^{-\frac{4}{3}(2mi^{-1/3}/5)^{3/2}}\leq ce^{-m/5}.$

Here we have used that $5-\kappa>2$, so $-\kappa i^{2/3}+mi^{-1/3}\geq
2mi^{-1/3}/5$. ∎

For a point process $\Pi$ on $\mathds{R}$, we say that a point $x\in\Pi$ is
$\delta$-jammed if there is a point $y\in\Pi$ such that $|x-y|\leq\delta$. We
will also need to bound the number of $\delta$-jammed points in the Airy point
process. We start with a proposition bounding the number of $\delta$-jammed
points in general determinantal processes.

######  Proposition 5.4.

Consider a deteminantal point process $\Pi$ on an interval $[a,a+\ell]$ with a
$C^{2}$ kernel $K$. Suppose that there exists a constant $b$ such that for all
$(x,y)\in[a,a+\ell]^{2}$,

$|K(x,y)|\leq b,\quad\left|\partial_{x}K(x,y)\right|\leq
b^{2},\quad\left|\partial_{y}K(x,y)\right|\leq
b^{2},\quad\;\text{and}\;\quad\left|\partial_{x,y}K(x,y)\right|\leq b^{3}.$

Let $L$ be the number of $\delta$-jammed points in $\Pi$. Then for every
$n\in\mathbb{N}$, we have that

$\mathbb{E}{\lfloor L/3\rfloor\choose n}n!\leq(4nb^{4}\delta^{3}\ell)^{n}.$

Here and throughout the remainder of the paper we use the convention that
${m\choose n}=0$ whenever $n>m$.

To prove Proposition 5.4, we start with a simple combinatorial lemma.

######  Lemma 5.5.

Let $\Pi$ be a point process on $[a,a+\ell]$ and let $L$ be the number of
$\delta$-jammed points in $\Pi$. Let $R$ be the number of $2n$-tuples
$(x_{1},y_{1},x_{2},y_{2},\dots,x_{n},y_{n})$ of distinct elements of
$[a,a+\ell]$ such that $|x_{i}-y_{i}|\leq\delta$ for all
$i\in\\{1,\dots,n\\}$. We have that

$R\geq{\lfloor L/3\rfloor\choose n}n!2^{n}.$

###### Proof.

Let $Z=\\{z_{1}<z_{2}<\dots<z_{L}\\}$ be the set of $\delta$-jammed points. We
can construct a partial matching on $Z$ via the following greedy algorithm. At
each step $i$, we match $z_{i}$ to $z_{i+1}$ if $z_{i}$ was not matched to
$z_{i-1}$ and if $|z_{i}-z_{i+1}|\leq\delta$. This algorithm produces a
matching with at least $\lfloor L/3\rfloor$ pairs $(z_{i},z_{i+1})$. By
counting the assignments of $(x_{i},y_{i})$ among the pairs in this matching,
we get the desired lower bound. ∎

###### Proof of Proposition 5.4.

Let $R$ be as in Lemma 5.5. We have that

$\mathbb{E}R=\int_{S}\det M,\qquad\text{where}\quad
S=\left\\{(x_{1},y_{1},\dots,x_{n},y_{n})\in[a,a+\ell]^{2n}:|x_{i}-y_{i}|\leq\delta\right\\}.$

In the above formula, $M$ is a $2n\times 2n$ matrix consisting of $n^{2}$
$2\times 2$ blocks, where each block is of the form

$M_{i,j}=\begin{bmatrix}K(x_{i},x_{j})&K(x_{i},y_{j})\\\
K(y_{i},x_{j})&K(y_{i},y_{j})\\\ \end{bmatrix}.$

We first bound $\det M$ on the set $S$. To do this, we will compute $\det
BMB^{t}$, where $B$ is a block diagonal matrix consisting of $n$ $2\times 2$
blocks of the form

$A=\begin{bmatrix}1/2&1/2\\\ (b\delta)^{-1}&-(b\delta)^{-1}\\\ \end{bmatrix}.$

We can calculate the blocks of $BMB^{t}$ by computing that

$A\begin{bmatrix}e&f\\\ g&h\\\
\end{bmatrix}A^{t}=\begin{bmatrix}\dfrac{e+f+g+h}{4}&\left(\dfrac{e-f}{\delta}+\dfrac{g-h}{\delta}\right)/2b\\\\[8.61108pt]
\left(\dfrac{e-g}{\delta}+\dfrac{f-h}{\delta}\right)/2b&\left(\dfrac{(e-g)}{\delta}-\dfrac{(f-h)}{\delta}\right)/(\delta
b^{2})\\\\[8.61108pt] \end{bmatrix}.$

Substituting in the entries of $M_{i,j}$ for $e,f,g,$ and $h$ above, we find
that the resulting $(1,1)$-entry is an average of values of $K$. The
$(1,2)$-entry is an average of difference quotients of $K$, multiplied by
$(\delta b)^{-1}(x_{j}-y_{j})$, and similarly the $(2,1)$-entry is an average
of difference quotients of $K$, multiplied by $(\delta b)^{-1}(x_{i}-y_{i})$.
Finally, the $(2,2)$-entry is a second difference quotient, multiplied by
$(\delta b)^{-2}(x_{i}-y_{i})(x_{j}-y_{j})$.

The mean value theorem then implies that all entries of $BMB^{t}$ are bounded
by $b$ on $S$ since $|x_{j}-y_{j}|\leq\delta$ for all $i\in\\{1,\dots,n\\}$ on
this set. Therefore by Hadamard’s inequality, we have that

$\det(M)=\frac{\det(BMB^{t})}{(\det
B)^{2}}=(b\delta)^{2n}\det(BMB^{t})\leq(b\delta)^{2n}\sqrt{2nb^{2}}^{\,2n}.$

Combining this inequality with the fact that the volume of $S$ is less than
$(2\delta\ell)^{n}$ shows that $\mathbb{E}R\leq(4nb^{4}\delta^{3}\ell)^{n}$.
Lemma 5.5 then completes the proof. ∎

######  Corollary 5.6.

There exists a constant $c$ such that for any $a\geq 1,\;\ell>0$, and
$n\in\mathbb{N}$, the number $L$ of $\delta$-jammed points in the Airy point
process in the interval $[-a,-a+\ell]$ satisfies

$\mathbb{E}{\lfloor L/3\rfloor\choose n}n!\leq(cna^{2}\delta^{3}\ell)^{n}.$

###### Proof.

Recall from (2) that we have the following formula for the Airy kernel:

$K(x,y)=\int_{0}^{\infty}{\operatorname{Ai}(x+\lambda)\operatorname{Ai}(y+\lambda)d\lambda}.$

To bound $K$ and its derivatives, we use the following bounds that hold for
all $x\geq 1$ (see [1], formulas 10.4.59-10.4.62).

$\displaystyle|\sqrt{x}\operatorname{Ai}(-x)|,\;|\operatorname{Ai}^{\prime}(-x)|\leq$
$\displaystyle\;cx^{1/4}\quad\;\text{and}\;\quad|\sqrt{x}\operatorname{Ai}(x)|,\;|\operatorname{Ai}^{\prime}(x)|\leq
cx^{1/4}e^{-(2/3)x^{3/2}}.$ (15)

Here $c$ is a positive constant. Also note that $\operatorname{Ai}(x)$ and
$\operatorname{Ai}^{\prime}(x)$ are bounded on $[-1,1]$ by continuity. Since
the Airy function is analytic and decays exponentially fast as $x\to\infty$,
we can differentiate under the integral sign to get formulas for
$\partial_{x}K,\partial_{y}K$ and $\partial_{x,y}K$. By the bounds in (15), we
get that

$|K(x,y)|\leq m^{1/2},\quad|\partial_{x}K(x,y)|\leq
m,\quad|\partial_{x,y}K(x,y)|\leq m^{3/2},$

where $m=c^{\prime}\max(-x,-y,1)$ for a constant $c^{\prime}$. Applying
Proposition 5.4 with $b=c^{\prime}a^{1/2}$ finishes the proof. ∎

As a consequence of Corollary 5.6, we can get the following tail bound on the
number of $\delta$-jammed points in the Airy point process in a given
interval.

######  Proposition 5.7.

There exists a constant $d>0$ such that for all $a\geq 1,\;\ell,m>0$, and
$\delta>0$, the number of $\delta$-jammed points $L$ in the Airy point process
in the interval $[-a,-a+\ell]$ satisfies

$\mathbb{P}(L>ma^{2}\delta^{3}\ell)\leq
2e^{-dm\left(\frac{a^{2}\delta^{3}\ell}{a^{2}\delta^{3}\ell+1}\right)}.$

###### Proof.

Observe that for every $n,m\in\mathbb{N}$ we have

$m^{n}\leq 4^{n}{{\left\lfloor m/3\right\rfloor}\choose n}n!+(12n)^{n}.$

Therefore by Corollary 5.6 and Fubini’s Theorem, for any $b>0$ we have that

$\mathbb{E}e^{bL}=\sum_{n=0}^{\infty}\frac{b^{n}\mathbb{E}L^{n}}{n!}\leq\sum_{n=0}^{\infty}\frac{(12nb)^{n}}{n!}+\frac{(4cna^{2}\delta^{3}\ell
b)^{n}}{n!}.$

Here the constant $c$ is as in Corollary 5.6. Therefore there is a universal
constant $d>0$ such that with $b=d/(a^{2}\delta^{3}\ell+1)$ we have
$\mathbb{E}e^{bL}\leq 2$. Applying Markov’s inequality to the event
$e^{bL}>e^{mba^{2}\delta^{3}\ell}$ completes the proof. ∎

In order to prove the bridge representation of the Airy line ensemble
$\mathcal{A}$, we first need to define associated graphs that record which
points in $\mathcal{A}$ are $\delta$-jammed.

######  Definition 5.8.

Fix $t>0$ and $\ell\in\mathbb{N}$. Define $s_{i}=it/\ell$ for all
$i\in\\{0,1,\dots,\ell\\}$. For $k,\delta>0$, we define a random graph
$G_{k}(t,\ell,\delta)$ on the set

$(\ell)=\\{1,\dots,k\\}\times\\{1,\dots,\ell\\},$

where the points $(i,j)$ and $(i+1,j)$ are connected if either

$|\mathcal{A}_{i}(s_{j-1})-\mathcal{A}_{i+1}(s_{j-1})|\leq\delta\quad\;\text{or}\;\quad|\mathcal{A}_{i}(s_{j})-\mathcal{A}_{i+1}(s_{j})|\leq\delta.$

Here $\mathcal{A}$ is the Airy line ensemble. Let $M_{k}(t,\ell,\delta)$ be
the size of the largest component of $G_{k}(t,\ell,\delta)$.

The second important consequence of Corollary 5.6 gives a bound on the size of
components in $G_{k}(t,\ell,\delta)$. In other words, it allows us to bound
the size of long chains of $\delta$-jammed points in the Airy line ensemble.

######  Proposition 5.9.

There exists $c>0$ so that for every $\delta\in(0,1]$, $\ell,m,k\in\mathbb{N}$
with $k\geq 2$ we have that

$\mathbb{P}(M_{k}(t,\ell,\delta k^{-1/3})\geq m)\leq\ell
k[cm\log(k/\delta)]^{m}\delta^{3{\left\lfloor m/6\right\rfloor}}.$ (16)

In the proof, $c$ is a constant that may change from line to line.

###### Proof.

Fix $m,k,\ell\in\mathbb{N}$ and $\delta>0$ with $k\geq 2$. We may assume that
$6\leq m\leq k$. We will work with the stationary Airy line ensemble
$\mathcal{R}$ instead of $\mathcal{A}$. This does not affect whether points
are $k^{-1/3}\delta$-jammed. If $M_{k}(t,\ell,\delta k^{-1/3})\geq m$, then
there must be some $(i,j)$ with $i\geq m$ in the set

$\partial S_{k}(\ell)=\\{1,\dots,k\\}\times\\{0,\dots,\ell\\}$

such that there are at least $m/2$ points in the set

$\\{\mathcal{R}_{i-m+1}(s_{j}),\dots,\mathcal{R}_{i-1}(s_{j}),\mathcal{R}_{i}(s_{j})\\}$

are $\delta k^{-1/3}$-jammed. Let $E_{k,b}$ be the event where

$|\mathcal{R}_{i}(s_{j})+\kappa i^{2/3}|<bi^{-1/3}\log(k+1)\quad\text{for
all}\quad(i,j)\in\partial S_{k}(\ell).$

On the event $E_{k,b}$, we have that

$\\{\mathcal{R}_{i-m+1}(s_{j}),\mathcal{R}_{i-1}(s_{j}),\dots,\mathcal{R}_{i}(s_{j})\\}\subset[-a_{i},-a_{i}+\ell_{i}]\quad\text{for
all}\quad(i,j)\in\partial S_{k}(\ell),$

where

$a_{i}=\kappa
i^{2/3}+bi^{-1/3}\log(k+1)\quad\;\text{and}\;\quad\ell_{i}=\kappa
mi^{-1/3}+2b(i-m+1)^{-1/3}\log(k+1).$

Therefore letting $L_{i,j}$ be the number of $\delta k^{-1/3}$-jammed in the
Airy point process $\\{\mathcal{R}_{r}(s_{j}):r\in\mathbb{N}\\}$ in the set
$[-a_{i},-a_{i}+\ell_{i}]$, a union bound implies that

$\mathbb{P}(M_{k}(t,\ell,\delta)\geq
m)\leq\mathbb{P}(E_{k}^{c})+\sum_{(i,j)\in\partial
S_{k}(\ell)}\mathbb{P}(L_{i,j}\geq m/2).$ (17)

Now set $b=\max(\beta,15\gamma{\left\lfloor m/6\right\rfloor})$, where $\beta$
is the constant from Corollary 5.3 and $\gamma=-\log_{k}\delta$. A union bound
implies that

$\mathbb{P}(E_{k}^{c})\leq c(\ell+1)k^{1-b/5}\leq c\ell
k^{1-3\gamma{\left\lfloor m/6\right\rfloor}}.$ (18)

It remains to bound $\mathbb{P}(L_{i,j}\geq m/2)$. By Markov’s inequality and
Corollary 5.6, we have that

$\mathbb{P}(L_{i,j}\geq m/2)\leq\mathbb{P}\left({{\left\lfloor
L_{i,j}/3\right\rfloor}\choose{\left\lfloor
m/6\right\rfloor}}>0\right)\leq\mathbb{E}{{\left\lfloor
L_{i,j}/3\right\rfloor}\choose{\left\lfloor m/6\right\rfloor}}{\left\lfloor
m/6\right\rfloor}!\leq(ca_{i}^{2}\delta^{3}k^{-1}\ell_{i})^{{\left\lfloor
m/6\right\rfloor}},$ (19)

where $c$ is a universal constant. In the first inequality, we used that
$m\geq 6$. Now, for a universal constant $c$, we have the bounds

$b\leq c(1+m\gamma)\quad\;\text{and}\;\quad(i-m+1)^{-1/3}\leq
i^{-1/3}+\frac{(m-1)}{i-1},$

which gives that

$\ell_{i}\leq
cm(1+\gamma)\left(i^{-1/3}+\frac{(m-1)}{i-1}\right)\log(k+1)\quad\;\text{and}\;\quad
a_{i}\leq\kappa i^{2/3}+c(1+m\gamma)i^{-1/3}\log(k+1).$

Using this we get that for every $i\in[m,k]\cap\mathbb{N}$, that

$a_{i}^{2}\ell_{i}\leq cm^{4/3}(1+\gamma^{3})k\log^{3}(k+1)$

The factor of $\gamma^{3}$ comes from the fact that there is a $b$ term in
each $a_{i}$ as well as $\ell_{i}$. Using that $\gamma=-\log_{k}\delta$, we
can bound the right hand side of (19) above by

$\left[cm^{4/3}\log^{3}(k/\delta)\delta^{3}\right]^{{\left\lfloor
m/6\right\rfloor}}.$

Combining this with the bound in (18) bounds the right hand side of (17). This
completes the proof. ∎

##  6 Preliminary modulus of continuity for Airy lines

In this section, we will obtain a modulus of continuity estimate for the Airy
line ensemble. This will be improved later as a consequence of the bridge
representation. However, we need to prove a preliminary estimate in order to
prove that theorem. When combined with the pointwise bounds in Corollary 5.3,
this modulus of continuity estimate gives a bound on the maximum of the $k$th
Airy line over an interval. This bound will be necessary for showing that
boundary conditions don’t propagate upwards when applying the Brownian Gibbs
property in a small region.

The main tools in the proof are stationarity and the fact that increments in
$\mathcal{A}_{k}$ dominate the bottom line of a Brownian $k$-melon. This leads
to moduli of continuity that are essentially sharp (up to a small
multiplicative constant) for the top lines, and increase proportionally to
$\sqrt{k}$.

First, we will need a tail bound on the difference between points in the Airy
line ensemble. The method of proof is a limiting version of the method used in
Proposition 4.1. It is cleaner than that proof because of stationarity.

######  Lemma 6.1.

There are constants $c,d>0$ such that for every
$t\in\mathds{R},s\in(0,1),a>0,$ and $k\in\mathbb{N}$ the $k$th line
$\mathcal{R}_{k}$ in the stationary Airy line ensemble satisfies

$\mathbb{P}(|\mathcal{R}_{k}(t)-\mathcal{R}_{k}(t+s)|>a\sqrt{s})\leq e^{ck-
da^{2}}.$ (20)

Moreover, for the top line $\mathcal{R}_{1}$ in the stationary Airy line
ensemble we have the stronger result that for every $t\in\mathds{R},s>0,$ and
$a>7s^{3/2},$ that

$\mathbb{P}(|\mathcal{R}_{1}(t)-\mathcal{R}_{1}(t+s)|>a\sqrt{s})\leq
ce^{-da^{2}}.$

###### Proof of Lemma 6.1.

For both statements, we may assume that $a>7s^{3/2}$, since by changing the
value of $c$ in (20), we can ensure that the right hand side is greater than
$1$ for smaller $a$. Set $b=a\sqrt{s}+s^{2}$. By stationarity of $\mathcal{R}$
and the equality in distribution when reversing time, we have that

$\displaystyle\mathbb{P}(|\mathcal{R}_{k}(t)-\mathcal{R}_{k}(t+s)|>a\sqrt{s})$
$\displaystyle=2\mathbb{P}(\mathcal{R}_{k}(0)-\mathcal{R}_{k}(s)>a\sqrt{s})$
$\displaystyle=2\mathbb{P}(\mathcal{A}_{k}(0)-\mathcal{A}_{k}(s)>b)$

For the final equality, we have replaced $\mathcal{R}$ by the non-stationary
version $\mathcal{A}$. Now set $r=b/(8s)$. Since $a>7s^{3/2}$, we have that
$r>s$. By the Brownian Gibbs property for $\mathcal{A}$, conditionally on the
set

$\\{\mathcal{A}_{i}(0),\mathcal{A}_{i}(r):i\in\\{1,\dots,k\\}\\}\cup\\{\mathcal{A}_{k+1}(q):q\in[0,r]\\},$

the restriction $\mathcal{A}|_{\\{1,\dots,k\\}\times[0,q]}$ is equal in
distribution to $k$ Brownian bridges $(\tilde{B}_{1},\dots,\tilde{B}_{k})$ on
$[0,r]$ of variance $2$ with endpoints given by $\mathcal{A}_{i}(0)$ and
$\mathcal{A}_{i}(r)$ for all $i\in[1,k]$, conditioned to avoid
$\mathcal{A}_{k+1}$. By Lemma 2.5, $(\tilde{B}_{1},\dots,\tilde{B}_{k})$
stochastically dominates the sequence

$(B_{1}+L,\dots,B_{k}+L),$

where $(B_{1},\dots,B_{k})$ is a Brownian $k$-melon with variance $2$ on
$[0,r]$, and $L$ is the linear function with $L(0)=\mathcal{A}_{k}(0)$ and
$L(r)=\mathcal{A}_{k}(r)$. We have that

$\displaystyle\mathbb{P}(\mathcal{A}_{k}(0)-\mathcal{A}_{k}(s)>b)$
$\displaystyle\leq\mathbb{P}([B_{k}+L](0)-[B_{k}+L](s)>b)$
$\displaystyle\leq\mathbb{P}\left(L(0)-L(s)\geq\frac{b}{2}\right)+\mathbb{P}\left(B_{k}(s)\leq-\frac{b}{2}\right).$

We can then relate the first term above to the values of $\mathcal{A}_{k}$ at
the times $0$ and $r=b/(8s)$, and then in turn to the values of
$\mathcal{A}_{k}$ at those times. This gives

$\displaystyle\mathbb{P}\left(L(0)-L(s)\geq\frac{b}{2}\right)$
$\displaystyle=\mathbb{P}\left(\mathcal{A}_{k}(0)-\mathcal{A}_{k}(r)\geq\frac{br}{2s}\right)$
$\displaystyle\leq\mathbb{P}\left(\mathcal{A}_{k}(r)\leq\kappa
k^{2/3}-\frac{br}{4s}\right)+\mathbb{P}\left(\mathcal{A}_{k}(0)\geq\kappa
k^{2/3}+\frac{br}{4s}\right)$

In terms of the stationary line ensemble, since $r=b/(2s)$ the above equals

$\mathbb{P}\left(\mathcal{R}_{k}(r)\leq\kappa
k^{2/3}-\frac{b^{2}}{64s^{2}}\right)+\mathbb{P}\left(\mathcal{R}_{k}(0)\geq\kappa
k^{2/3}+\frac{b^{2}}{32s^{2}}\right).$

We can bound the two probabilities above using Corollary 5.3. Also,
$\mathbb{P}\left(B_{k}(s)\leq-b/2\right)$ can be bounded by Lemma 3.2.
Combining these bounds gives that

$\mathbb{P}(\mathcal{A}_{k}(0)-\mathcal{A}_{k}(s)>b)\leq
ce^{-db^{2}/s^{2}}+e^{ck-db^{2}/s}$

for constants $c$ and $d$. Using that $b\geq a\sqrt{s}$ and that $s\in(0,1)$
then proves the first part of the lemma. For the second part of the lemma, we
can use the usual Tracy-Widom tail bounds instead of Corollary 5.3 and the
fact that $a>7s^{3/2}$ to give the improved bound. ∎

By combining Lemma 3.3 and Lemma 6.1 we get the following modulus of
continuity for the Airy line ensemble.

######  Theorem 6.2.

There are $c,d>0$ so that for any $k\in\mathbb{N}$, $a\in\mathds{R}$, there
exists a random constant $C_{k}$ so that

$|\mathcal{R}_{k}(t)-\mathcal{R}_{k}(t+s)|\leq C_{k}\sqrt{s}\log^{1/2}(2/s)$

for all $t,t+s\in[a,a+1]$. Moreover

$\mathbb{P}(C_{k}>m)\leq e^{ck-dm^{2}}\qquad\text{for all }m>0.$

We record the the following corollary of Theorem 6.2, which confines lines in
the stationary Airy line ensemble.

######  Corollary 6.3.

There exists $\beta,c>0$ such that for any $k\in\mathbb{N}$, any $t>0$, and
any $m>\beta\log(k+1)$, we have that

$\mathbb{P}\left(\sup_{s\in[0,t]}|\mathcal{R}_{k}(s)+\kappa
k^{2/3}|>mk^{-1/3}\right)\leq(t+1)ce^{-m/10}.$

###### Proof.

It suffices to proves the statement for $t\leq 1$, as for larger $t$, we can
simply use a union bound and the stationarity of $\mathcal{R}_{k}$. Let
$\Pi=\\{0,t/k^{2},2t/k^{2},\dots,t\\}$. Letting $C_{k}$ be as in Theorem 6.2
for the interval $[0,t]$, we have that

$\sup_{s\in[0,t]}|\mathcal{R}_{k}(s)+\kappa
k^{2/3}|\leq\sup_{s\in\Pi}|\mathcal{R}_{k}(s)+\kappa
k^{2/3}|+C_{k}\frac{\sqrt{t}}{k}\log^{1/2}(2k^{2}).$

The probability that the first term on the right hand side above is large can
be bounded by using Corollary 5.3 and a union bound. The probability that the
second term is large can be bounded by Theorem 6.2. Combining these proves the
lemma. ∎

##  7 The bridge representation

In this section we prove the bridge representation, Theorem 1.1. After proving
the bridge representation, we prove a small proposition that shows that at
most locations, the bridge representation samples bridges without any non-
intersection condition.

We begin with a definition. Let $t>0,\ell\in\mathbb{N}$ and let
$s_{j}=jt/\ell$ as in Section 5. Recall that $G_{k}(t,\ell,\delta)$ is a
random graph on the set $\\{1,\dots,k\\}\times\\{1,\dots,\ell\\}$ where
vertices $(i,j)$ and $(i+1,j)$ are connected if either

$|\mathcal{A}_{i}(s_{j-1})-\mathcal{A}_{i+1}(s_{j-1})|\leq\delta\qquad\;\text{or}\;\qquad|\mathcal{A}_{i}(s_{j})-\mathcal{A}_{i+1}(s_{j})|\leq\delta.$

######  Definition 7.1.

The bridge representation $\mathcal{B}^{k}(t,\ell,\delta)$ of the Airy line
ensemble $\mathcal{A}$ is an ordered line ensemble
$(\mathcal{B}^{k}_{1},\dots,\mathcal{B}^{k}_{2k})$ with domain $[0,t]$
constructed as follows. For every
$(i,j)\in\\{1,\dots,2k\\}\times\\{1,\dots,\ell\\}$, sample a Brownian bridge
$B_{i,j}:[s_{j-1},s_{j}]\to\mathds{R}$ with

$B_{i,j}(s_{j-1})=\mathcal{A}_{i}(s_{j-1})\quad\;\text{and}\;\quad
B_{i,j}(s_{j})=\mathcal{A}_{i}(s_{j}),$

where the bridges $B_{i,j}$ and $B_{i^{\prime},j}$ are conditioned not to
intersect if $(i,j)$ and $(i^{\prime},j)$ are in the same component of
$G_{2k}(t,\ell,\delta)$. We then define the $i$th line $\mathcal{B}^{k}_{i}$
of the line ensemble $\mathcal{B}^{k}(t,\ell,\delta)$ by concatenating the
bridges $B_{i,j}$. That is, $\mathcal{B}^{k}_{i}|_{[s_{j-1},s_{j}]}=B_{i,j}$
for all $j\in\\{1,\dots,\ell\\}$.

We can now state the main theorem of this paper, which shows that
$\mathcal{B}$ and $\mathcal{L}$ are close in law when restricted to the set
$\\{1,\dots,k\\}\times[0,t]$ in the appropriate parameter range.

######  Theorem 7.2.

There exist constants $c,d>0$ such that the following holds for all $k\geq 3$,
$t>0$, $\gamma\in(c\log(\log k)/\log k,2]$ and $\ell\geq tk^{2/3+\gamma}$. The
total variation distance between the laws of
$\mathcal{B}^{k}(t,\ell,k^{-1/3-\gamma/4})|_{\\{1,\dots,k\\}\times[0,t]}$ and
$\mathcal{A}|_{\\{1,\dots,k\\}\times[0,t]}$ is bounded above by

$\ell e^{-d\gamma k^{\gamma/12}}.$

Theorem 7.2 can be thought of as giving a quantitative version of the Brownian
Gibbs property for the Airy line ensemble. In particular, it allows us to
apply the Brownian Gibbs property to large patches in the Airy line ensemble
without having to worry about boundary conditions or long-range interactions
between lines (note that the components in
$G_{k}(t_{k},\ell_{k},k^{-1/3-\gamma/4})$ are of bounded size as we take
$k\to\infty$ by Proposition 5.9 as long as the growth of $\ell_{k}$ is bounded
by $k^{a}$ for some $a$).

Note that the upper bound on $\gamma$ in Theorem 7.2 is quite artificial and
is chosen purely for the sake of proof. For any $\gamma>c\log\log(k)/\log k$,
we can get a similar total variation bound, though the exponent $\gamma/12$
will be replaced by $b\gamma$ for a constant $b$ that decays as
$\gamma\to\infty$. The restriction that $k\geq 3$ is there purely to ensure
that $\log\log k/\log k>0$.

The first step in the proof of Theorem 7.2 is to show that the lower boundary
condition becomes only affects nearby lines when we apply the Brownian Gibbs
property to $\mathcal{A}$ on a region of the form
$\\{1,\dots,k\\}\times[0,k^{-2/3-\gamma}]$.

######  Lemma 7.3.

Define the $\sigma$-fields

$\displaystyle\mathcal{F}$
$\displaystyle=\sigma\\{\mathcal{A}_{i}(s):(i,s)\in\\{1,\dots,2k\\}\times\\{0,t\\}\;\text{
or }\;i=2k+1,s\in[0,t]\\}\quad\;\text{and}\;$ $\displaystyle\mathcal{G}$
$\displaystyle=\sigma\\{\mathcal{A}_{i}(s):(i,s)\in\\{1,\dots,2k\\}\times\\{0,t\\}\\}.$

There exist a $c>0$ such that for every $k\geq 3$, $\gamma\in[c\log\log k/\log
k,2]$ and $t\in(0,k^{-2/3-\gamma}]$, there exists a $\mathcal{G}$-measurable
random variable $J\in(k,2k]$ such that the following holds.

Let $A=(A_{J},\dots,A_{2k})$ be a sequence of $2k+1-J$ random functions, such
that conditionally on $\mathcal{F}(k,t)$, the sequence $A$ consists of
$2k+1-J$ independent Brownian bridges of variance $2$ with endpoints
$A_{i}(0)=\mathcal{A}_{i}(0)$ and $A_{i}(t)=\mathcal{A}_{i}(t)$ conditioned
not to intersect each other or the lower boundary $\mathcal{A}_{2k+1}$. Then

$\mathbb{P}\Big{(}\min_{s\in\\{0,t\\}}\mathcal{A}_{J-1}(s)-\max_{r\in[0,t]}A_{J}(r)\geq
k^{-1/3-\gamma/4}\;\;\Big{|}\;\mathcal{F}\Big{)}\geq 1-e^{-d\gamma
k^{\gamma/4}}$ (21)

with probability at least $1-e^{-d\gamma k^{\gamma/12}}$. Here $d$ is a
universal constant.

See Figure 3 for an illustration of the lemma and the main idea of the proof.

(a)

(b)

Figure 3: Lemma 7.3 essentially shows that on the right scale, it is highly
unlikely that the effect of the lower boundary condition is propagated to the
top $k$ lines when applying the Brownian Gibbs property on the top $2k$ lines.
More precisely, there exists a $J>k$ (the level of the top blue points) for
which the bridge connecting the points at level $J$ stays well below the two
black boundary points at level $J-1$. This is illustrated in Figure 3(a).
Figure 3(b) illustrates how this is proven. We shift some blue points up so
that with high probability, independent Brownian bridges between all the blue
points will not intersect each other or the lower boundary. This shifting
process stabilizes by the point $J$. Therefore we can conclude that the bridge
at level $J$ is stochastically dominated by a Brownian bridge off of a low
probability event.

###### Proof.

Fix $k\in\mathbb{N}$ and $\gamma\in[c\log\log k/\log k,2]$ for some large
constant $c$. Exactly how large we need to take $c$ for the lemma to hold will
be made clear in the proof. Let $m\in\mathbb{N}$ be such that $m<k/4$, and let
$b\in(0,k^{\gamma/4}/m]$. Set $a_{i}(s)=-\kappa i^{2/3}-s^{2}$ (an
approximation to $\mathbb{E}\mathcal{A}_{i}(s)$). Let $E_{b,m}$ be the event
where:

  1. (i)

$\mathcal{A}_{2k+1}(s)\leq a_{2k+1}(s)+bk^{-1/3}$ for all $s\in[0,t]$.

  2. (ii)

$\mathcal{A}_{\left\lfloor 3k/2\right\rfloor}(s)\geq a_{\left\lfloor
3k/2\right\rfloor}(s)-bk^{-1/3}$ for $s=0,t$.

  3. (iii)

The graph $G_{2k}(t,1,5k^{-1/3-\gamma/4})$ has no component of size greater
than $m$.

We define

$J=\max\Big{\\{}j\leq\frac{3k}{2}:\min_{s\in\\{0,t\\}}\mathcal{A}_{j-1}(s)-\mathcal{A}_{j}(s)\geq
4k^{-1/3-\gamma/4}\Big{\\}}.$

On the event $E_{b,m}$, condition (iii) implies that $J>k$. We now bound the
left hand side of (21) for this $J$ on the event $E_{b,m}$.

For this, we set up a stochastic domination argument. To begin, we would like
to find points $h_{i}$ for $i=1,\ldots,2k$ that dominate $\mathcal{A}_{i}(0)$
and have spacing at least $b\sqrt{t}$. First define
$h_{2k+1}=a_{2k+1}(0)+bk^{-1/3}$ . For $i\leq 2k$, recursively define

$h_{i-1}=\max(h_{i}+b\sqrt{t},\mathcal{A}_{i-1}(0)).$

We first claim that

$h_{J}\leq\mathcal{A}_{J-1}(0)-3k^{-1/3-\gamma/4}$ (22)

on the event $E_{b,m}$. To show this, set

$I=\\{i\in[J,2k]:h_{i}=\mathcal{A}_{i}(0)\\}.$

We first claim that $\sup I\geq 3k/2$. If $\sup I<3k/2$, then

$\begin{split}h_{{\left\lfloor
3k/2\right\rfloor}}&=a_{2k+1}(s)+bk^{-1/3}+\left(2k+1-{\left\lfloor
3k/2\right\rfloor}\right)\sqrt{t}b\\\ &\leq
a_{2k+1}(s)+k^{-1/3+\gamma/4}+k^{2/3-\gamma/4}.\end{split}$ (23)

Here the inequality uses that $t\leq k^{-2/3-\gamma}$ and $b\leq
k^{\gamma/4}$. However, by (ii) in the definition of $E_{b,m}$, again using
that $b\leq k^{\gamma/4}$, we have that

$h_{{\left\lfloor 3k/2\right\rfloor}}\geq a_{{\left\lfloor
3k/2\right\rfloor}}(s)-k^{\gamma/4-1/3}.$

This bound contradicts (23) if we can show that

$k^{-1/3+\gamma/4}+k^{2/3-\gamma/4}+k^{\gamma/4-1/3}\leq\kappa(2^{2/3}-(3/2)^{2/3})k^{2/3}.$

Since $\gamma\leq 2$, the right hand side above is bounded by
$3k^{2/3-\gamma/4}$. The assumption that $\gamma>c\log(\log k)/\log k$ then
implies the above as long as $c$ is taken large enough. Now, for $i\leq\sup
I$, the recursive definition of $h_{i}$ implies that

$h_{i}-\mathcal{A}_{i}(0)=[\inf\\{j\geq
i:h_{j}=\mathcal{A}_{j}(0)\\}-i]b\sqrt{t}.$

By condition (iii) on the set $E_{b,m}$ this implies that

$h_{i}-\mathcal{A}_{i}(0)\leq mb\sqrt{t}\leq k^{-1/3-\gamma/4}.$

In particular, this holds for $i=J$. We can define $h_{i}^{\prime}$ in terms
of the sequence $\\{\mathcal{A}_{i}(t)\\}$ analogously to how we defined
$h_{i}$. This gives an analogous inequality for $h^{\prime}_{J}$.

Now, by Lemma 2.5, on the event $E_{b,m}$ the distribution of
$\\{A_{J}(s):s\in[0,t]\\}$ conditional on $\mathcal{F}(k,t)$ is stochastically
dominated by the top line $B_{J}$ of a system of Brownian bridges
$(B_{J},\ldots,B_{2k})$ of variance $2$ on $[0,t]$ with endpoints
$B_{i}(0)=h_{i},B_{i}(t)=h^{\prime}_{i}$, conditioned to avoid each other and
stay above the boundary

$h_{2k+1}(s)=a_{2k+1}(s)+bk^{-1/3}\log k.$

In particular on $E_{b,m}$ we can bound the conditional probability in (21)
below by

$\mathbb{P}\Big{(}\min_{s\in\\{0,t\\}}\mathcal{A}_{J-1}(s)-\max_{r\in[0,t]}B_{J}(r)\geq
k^{-1/3-\gamma/4}\;\Big{|}\;\mathcal{F}\Big{)}$ (24)

Now the $B_{i}$ can be realized by repeated sampling of independent Brownian
bridges until the avoidance conditions are satisfied. Let
$(C_{J},\dots,C_{2k})$ denote the first sample, and let $S$ be the event that
it successfully satisfies all avoidance conditions. Then the event in (24) is
implied by

$S\cap\left\\{\min_{s\in\\{0,t\\}}\mathcal{A}_{J-1}(s)-\max_{r\in[0,t]}C_{J}(r)\geq
k^{-1/3-\gamma/4}\right\\}.$

By construction, this event is implied by all bridges $C_{i}$ staying in a
channel of width $b\sqrt{t}/2$ about their mean. By a standard bound on the
maximum of a Brownian bridge, on the event $E_{b,m}$ the conditional
probability in (21) bounded below by

$1-cke^{-db^{2}}$ (25)

for universal constants $c$ and $d$. We can lower bound the probability of the
event $E_{b,m}$ by using Corollary 6.3 for condition (i), Corollary 5.3 for
condition (ii), and Proposition 5.9 for condition (iii). For large enough $k$,
these bounds give that

$\mathbb{P}(E_{b,m})\geq 1-ce^{-b/10}-[c(1+\gamma/4)m\log
k]^{m}k^{1-\frac{3\gamma}{4}{\left\lfloor m/6\right\rfloor}}$

for a constant $c$. Taking $b=k^{\gamma/6}$ and $m=k^{\gamma/12}$ and using
that $\gamma\in[c\log\log k/\log k,2]$ gives that the right hand side above is
bounded below by $1-e^{d\gamma k^{\gamma/12}}$ for a constant $d$. Moreover,
the assumption on $\gamma$ also implies that (25) is bounded below by
$1-e^{-dk^{\gamma/4}}$. ∎

We are now ready to prove the bridge representation theorem. We use the
notation of Lemma 7.3. Throughout the proof, $b$ and $d$ will be universal
constants that may change from line to line.

###### Proof of Theorem 7.2.

Set $\gamma\in[c\log\log k/\log k,2]$ for a sufficiently large constant $c$.
As with Lemma 7.3, exactly how large we need to take $c$ will be made clear in
the proof. Let $\mathcal{H}$ be the $\sigma$-algebra generated by the random
variables

$\\{\mathcal{A}_{i}(s_{j}):(i,j)\in\\{1,\dots,2k\\}\times\\{0,\dots,\ell\\}\\}\;\;\text{and
the boundary}\;\;\\{\mathcal{A}_{2k+1}(s):s\in[0,t]\\}.$

By Lemma 7.3 and a union bound, there exist random variables
$J_{1},\dots,J_{\ell}\in(k,2k]$ such that the following holds. For
$j\in\\{1,\dots,\ell\\}$, define a sequence of Brownian bridges
$A_{j}=(A_{J_{i},j},\dots,A_{2k,j})$ on the interval $[s_{j-1},s_{j}]$ as in
the statement of Lemma 7.3 such that
$A_{i,j}(s_{j-1})=\mathcal{A}_{i}(s_{j-1})$ and
$A_{i,j}(s_{j})=\mathcal{A}_{i}(s_{j})$, and such that the bridges in $A_{j}$
are conditioned to avoid each other and the boundary $\mathcal{A}_{2k+1}$.
Then with probability at least $1-\ell e^{-dk^{\gamma/12}}$, we have

$\begin{split}\mathbb{P}\Bigg{(}\min_{i\in\\{1,\dots,\ell\\}}\left\\{\min_{q\in\\{s_{i},s_{i-1}\\}}\mathcal{A}_{J_{i}-1}(q)-\max_{r\in[s_{i-1},s_{i}]}A_{i,J_{i}}(r)\right\\}\geq
k^{-1/3-\gamma/4}\;\Bigg{|}\;\mathcal{H}\Bigg{)}\geq 1-\ell e^{-d\gamma
k^{\gamma/4}}.\end{split}$ (26)

We now define a line ensemble $\mathcal{C}^{k}$ on the set
$\\{1,\dots,2k\\}\times[0,t]$.

For each $i\in\\{J_{i},\dots,2k\\}$ and $j\in\\{1,\dots,\ell\\}$, set
$C_{i,j}=A_{i,j}$. For $i\in\\{1,\dots,J_{i}-1\\}$ and
$j\in\\{1,\dots,\ell\\}$, set $C_{i,j}=B_{i,j}$, where the bridges $B_{i,j}$
are as in Definition 7.1. Concatenating the bridges $C_{i,j}$ together for
$j\in\\{1,\dots,\ell\\}$ gives the $i$th line $\mathcal{C}^{k}_{i}$ in the
line ensemble $\mathcal{C}^{k}$.

Now, the conditional distributions of
$\mathcal{A}|_{\\{1,\dots,2k\\}\times[0,t]}$ and $\mathcal{C}^{k}$, given
$\mathcal{H}$, are given by the laws of $2k\ell$ independent Brownian bridges

$\\{Z_{i,j}:(i,j)\in\\{1,\dots,2k\\}\times\\{1,\dots,\ell\\}\\}$

with endpoints $Z_{i,j}(s_{j-1})=\mathcal{A}_{i}(s_{j-1})$ and
$Z_{i,j}(s_{j})=\mathcal{A}_{i}(s_{j})$, restricted to
$\mathcal{H}$-measurable sets given by the avoidance conditions on
$\mathcal{A}$ and $\mathcal{C}^{k}$. Since the avoidance conditions for
$\mathcal{A}$ are stronger than those of $\mathcal{C}$, the total variation
distance of the laws is simply the probability that the avoidance conditions
for $\mathcal{A}$ fail, given that the conditions for $\mathcal{C}^{k}$
succeed. This is equal to

$\mathbb{P}(\text{$\mathcal{C}^{k}$ intersects itself or the lower boundary
$\mathcal{A}_{2k+1}$}).$

We now bound the probability that the line ensemble $\mathcal{C}^{k}$ has an
intersection on the set $E$ where (26) holds. This can be bounded above by the
probability that for some $(i,j)$ with $j<J_{i}$, one of the bridges $C_{i,j}$
leaves a channel of width $k^{-1/3-\gamma/4}/2$ around the line between its
endpoints. Letting $M$ be the size of the largest component of
$G_{2k}(t,\ell,\delta)$, by Proposition 3.4 and a union bound, the conditional
probability of this event given $\mathcal{H}$ is bounded above by

$b\ell ke^{-dk^{\gamma/2}M^{-1}}.$

Here we have used the fact that the width of each of the intervals
$[s_{j-1},s_{j}]$ is bounded above by $k^{-2/3-\gamma}$. By Proposition 5.9,
we also have that

$\mathbb{P}(M>m)\leq\ell[c(1+\gamma^{3})m\log
k]^{m}k^{1-3(\gamma/4){\left\lfloor m/6\right\rfloor}},$

and so plugging in the case $m=k^{\gamma/12}$, we get that

$\mathbb{P}\left(c\ell ke^{-dk^{\gamma/2}M^{-1}}>c\ell
ke^{-dk^{\gamma/4}}\right)\leq\ell e^{-d\gamma k^{\gamma/12}}.$

Here we have used that $\gamma\in[c\log\log k/\log k,2]$ to simplify the upper
bound. Hence on the event $E$, we get that

$\mathbb{P}(\text{$\mathcal{C}_{k}$ intersects itself or the lower boundary
$\mathcal{A}_{2k+1}$}|E)\leq\ell e^{-d\gamma k^{\gamma/12}}+b\ell
ke^{-dk^{\gamma/4}}\leq\ell e^{-d\gamma k^{\gamma/12}}.$

Again, the last inequality follows since $\gamma\geq c\log\log k/\log k$.
Using the bound in (26), we also get that $\mathbb{P}(E^{c})\leq\ell
e^{-d\gamma k^{\gamma/12}}.$ Combining this with the above bound then
completes the proof. ∎

We conclude this section by providing an estimate on the number of edges in
the graph $G_{2k}(t,\ell,\delta)$. We also show that these edges are evenly
spread out. Note that already by Proposition 5.9, we have good control over
the size of the components in the graph $G_{2k}(t,\ell,\delta)$.

######  Proposition 7.4.

Fix $\gamma>0$, and let $k\in\mathbb{N},t\in[0,\infty),\ell\geq
tk^{2/3+\gamma}$. Let the graph $G_{2k}(t,\ell,k^{-1/3-\gamma/4})$ be as in
Definition 5.8. For each $j\in\\{1,\dots,\ell\\}$, let

$V_{j}=\\{x\in\\{1,\dots,2k\\}:\deg_{G}(x,j)\geq 1\\}.$

In other words, $V_{j}$ is the set of vertices in $G$ with second coordinate
$j$ that are connected to at least one other vertex. Then for any
$\alpha\in(0,1],\;i\in\\{{\left\lfloor
k^{\alpha}\right\rfloor}+1,\dots,2k\\},$ and $m\leq k^{2\alpha/3}$, we have
that

$\mathbb{P}(|V_{j}\cap\\{i-{\left\lfloor
k^{\alpha}\right\rfloor},\dots,i\\}|>mk^{\alpha-3\gamma/4})\leq ce^{-dm}.$

Here $c$ and $d$ are universal constants.

In the above lemma, we have not tried to optimize the bounds. In particular,
the restriction on $m$ is rather artificial, and is chosen for ease of proof.
Rather, we have just tried to demonstrate how the edges in the graph
$G_{2k}(t,\ell,k^{-1/3-\gamma/4})$ can be controlled.

###### Proof.

For notational convenience, we will work with the stationary Airy line
ensemble $\mathcal{R}$. By Corollary 5.3, the following holds for any
$s\in\mathds{R}$ and any $i,k\in\mathbb{N}$.

$\displaystyle\mathbb{P}$
$\displaystyle\Big{(}\\{\mathcal{R}_{i}(s),\dots,\mathcal{R}_{i+{\left\lfloor
k^{\alpha}\right\rfloor}}(s)\\}\subset I\Big{)}\geq
1-2e^{-k^{2\alpha/3}/5},\qquad\text{where}$ (27) $\displaystyle I$
$\displaystyle=[-c_{0}(i+k^{\alpha})^{2/3}-k^{2\alpha/3}i^{-1/3},-c_{0}i^{2/3}+k^{2\alpha/3}i^{-1/3}].$

The interval $I$ above is of the form $[-a,-a+\ell]$ where the product
$a^{2}\ell\leq ck^{1+\alpha}$, for a constant $c$ that does not depend on any
of $\alpha,i$ and $k$. Here we have used the assumption that $m\leq
k^{2\alpha/3}$. Therefore by Proposition 5.7, the number $L$ of
$k^{-1/3-\gamma/4}$-jammed points in the interval $I$ satisfies

$\mathbb{P}(L>mk^{\alpha-3\gamma/4})\leq ce^{-dm}.$

for universal constants $c$ and $d$. By combining this with the bound in (27),
we can bound the size of $V_{j}\cap\\{i-{\left\lfloor
k^{\alpha}\right\rfloor},\dots,i\\})$ by noting that the number of non-
isolated vertices can be bounded by the sum of the number of $\delta$-jammed
points in the sequences $\\{\mathcal{R}_{i}(s_{j-1})\\}_{i\in\mathbb{N}}$ and
$\\{\mathcal{R}_{i}(s_{j})\\}_{i\in\mathbb{N}}$. ∎

##  8 A stronger modulus of continuity for Airy lines

In this section we give an application of the bridge representation to get a
stronger modulus of continuity bound on sets of Airy lines than the one given
in Section 6.

######  Theorem 8.1.

There exists a constant $d>0$ such that for any $t>0$, we have

$\sum_{k\in\mathbb{N}}\mathbb{P}\left(\sup_{s,s+r\in[0,t]}\frac{|\mathcal{A}_{k}(s)-\mathcal{A}_{k}(s+r)|}{\sqrt{r}\log^{1/2}(1+r^{-1})\log^{d}k}>1\right)<\infty.$
(28)

In particular, for any $t>0$, we have that

$\sup_{k\in\mathbb{N}}\sup_{s,s+r\in[0,t]}\frac{|\mathcal{A}_{k}(s)-\mathcal{A}_{k}(s+r)|}{\sqrt{r}\log^{1/2}(1+r^{-1})\log^{d}k}<\infty\qquad\text{
almost surely }.$

The second statement follows from the first by the Borel-Cantelli lemma and
Theorem 6.2. Note that if we remove the supremum over $k$, then the second
statement is implied by Theorem 6.2. However, if we simply apply that theorem
and take a union bound, we need to put a factor of $\sqrt{k}$ in the
denominator, rather than a power of $\log(k)$. Note also that on small time
intervals (i.e. when $r<t^{-2/3-\gamma}$ for fixed $\gamma>0$), a variant of
our proof will give the power $\log^{1/2}(k)$, which is the same as what one
would get for sequences of independent Brownian motions.

Throughout the proof, $b$ and $d$ are constants that may change from line to
line.

###### Proof.

We will prove the statement with a $t$-dependent constant $b(t)$ in place of
$1$ in (28); we can later switch $b(t)$ to be equal to $1$ by increasing $d$
by $1$. It suffices to prove this modified statement for $t=1$, as we can use
a union bound and the triangle inequality (at the expense of increasing
$b(t)$) to get the same statement for larger $t$. The parabolic shape of
$\mathcal{A}$ will affect differences for larger $t$, but this can again be
dealt with by increasing $b(t)$. Let $\mathcal{B}^{k}$ denote the bridge
representation

$\mathcal{B}^{k}\left(1,{\left\lceil
k^{2/3+\gamma}\right\rceil},k^{-1/3-\gamma/4}\right)$

with $\gamma=c\log\log(k)/\log(k)$. Here $c$ is a universal constant that can
be chosen to be large enough so that each of the subsequent steps in the proof
go through. By Theorem 7.2, for all $k$ such that $c\log\log(k)/\log(k)<2$, we
can couple the bridge representations $\mathcal{B}^{k}$ to the Airy line
ensemble $\mathcal{A}$ so that

$\mathbb{P}(\mathcal{B}^{k}|_{\\{1,\dots,k\\}\times[0,1]}\neq\mathcal{A}|_{\\{1,\dots,k\\}\times[0,1]})\leq{\left\lceil
k^{2/3+\gamma}\right\rceil}e^{-d\gamma k^{\gamma/12}}.$

By noting that $k^{\gamma}=\log^{c}k$, as long as $c$ was chosen large enough,
the right hand side above is bounded by $k^{-2}$ for all large enough $k$ and
so

$\sum_{k\in\mathbb{N}}\mathbb{P}(\mathcal{B}^{k}|_{\\{1,\dots,k\\}\times[0,1]}\neq\mathcal{A}|_{\\{1,\dots,k\\}\times[0,1]})<\infty.$

In particular, this means that it suffices to prove (28) with the $k$th line
of $\mathcal{B}^{k}$, denoted by $B_{k}$, in place of $\mathcal{A}_{k}$.
Specifically, it is enough to show that for some constants $b,d>0$, that

$\mathbb{P}\left(\sup_{s,s+r\in[0,1]}\frac{|B_{k}(s)-B_{k}(s+r)|}{\sqrt{r}\log^{1/2}(1+r^{-1})\log^{d}k}>b\right)=O(k^{-2}).$
(29)

Define

$s_{i}=\frac{i}{{\left\lceil
k^{2/3+\gamma}\right\rceil}}\quad\;\text{for}\;\quad
i\in\left\\{0,\dots,{\left\lceil k^{2/3+\gamma}\right\rceil}\right\\}.$

Let $L_{k}$ denote the function which is equal to $B_{k}$ at the grid points
$s_{i}$ and linear on each of the intervals $[s_{i-1},s_{i}]$ and set
$X_{k}=B_{k}-L_{k}$. It suffices to prove the bound in (29) for $L_{k}$ and
$X_{k}$ separately. Letting $a_{k}(s)=\kappa k^{2/3}-s^{2}$, Corollary 5.3 and
a union bound implies that for some constant $d>0$,

$\mathbb{P}\left(|B_{k}(s_{i})+a_{k}(s_{i})|>m\log(k)k^{-1/3}\text{ for some
}i\in\left\\{0,\dots,{\left\lceil
k^{2/3+\gamma}\right\rceil}\right\\}\right)\leq e^{(d-m)\log k/5}$

Hence there exists a constant $b>0$ that is independent of $k$ for which

$\mathbb{P}\left(|L_{k}(s)-L_{k}(s+r)|\leq b\log(k)k^{-1/3}\max(r{\left\lceil
k^{2/3+\gamma}\right\rceil},1)\text{ for all }s,s+r\in[0,1]\right)\geq
1-bk^{-2}.$ (30)

Since $k^{\gamma}=\log^{c}(k)$, we have that

$b\log(k)k^{-1/3}\max(r{\left\lceil
k^{2/3+\gamma}\right\rceil},1)\leq\log^{c+2}(k)\sqrt{r},$

for all large enough $k$. This implies the bound in (29) for $L_{k}$.

To get a modulus of continuity bound on $X_{k}$, we use the modulus of
continuity bounds on Brownian $k$-melons established in Proposition 3.4. First
we need to bound the size of components in the underlying graph
$G^{k}=G_{2k}(1,{\left\lceil k^{2/3+\gamma}\right\rceil},k^{-1/3-\gamma/4})$
that gives rise to the bridge representation $\mathcal{B}^{k}$. Letting
$M^{k}$ be the size of the largest component in $G^{k}$, Proposition 5.9
implies that there exists a constant $b>0$ such that for every $m>0$ and
$k\in\mathbb{N}$, we have

$\mathbb{P}(M_{k}>m)\leq
k^{2}[bm\log(k)]^{m}[\log(k)]^{-\frac{3c}{4}{\left\lfloor m/6\right\rfloor}}.$

To get this bound we have again used the observation that
$k^{\gamma}=\log^{c}k$, as well as the crude bound that ${\left\lceil
k^{2/3+\gamma}\right\rceil}\leq k$ for large enough $k$. In particular,
setting $m=\log^{c/16}k$, as long as $c$ is large enough we get that

$\mathbb{P}(M_{k}>\log^{c/16}k)\leq k^{-2}$ (31)

for all large enough $k$. Let $F_{k}$ denote the event where
$M_{k}\leq\log^{c/16}k$. Proposition 3.4 and a union bound implies that there
exists a constant $b>0$ such that

$\mathbb{P}\left(\sup_{s,s+r\in[0,1]}\frac{|X_{k}(s)-X_{k}(s+r)|}{\log^{1+c/32}(k)\sqrt{r}\log^{1/2}(1/r)}>b\;\bigg{|}\;\;F_{k}\right)\leq
bk^{-2}.$

Combining this with (31) gives the bound in (29) for $X_{k}$, completing the
proof. ∎

Acknowledgments. D.D. was supported by an NSERC CGS D scholarship. B.V. was
supported by the Canada Research Chair program, the NSERC Discovery
Accelerator grant, the MTA Momentum Random Spectra research group, and the ERC
consolidator grant 648017 (Abert). B.V. thanks Janosch Ortmann for several
useful discussions.

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