# On the association of the Airy1 process

Fei Pu

###### Abstract

We first show that the Airy1 process is associated using the association
property of solution to stochastic heat equation and convergence of the KPZ
equation to the KPZ fixed point. Then we apply Newman’s inequality to
establish the ergodicity and central limit theorem for the Airy1 process.
Combined with the asymptotic behavior of the tail probability, we derive a
Poisson limit theorem for the Airy1 process and give a precise estimate on the
asymptotic behavior of the maximum of the the Airy1 process over an interval.
Analogous results for the Airy2 process are also presented.

MSC 2010 subject classification: 60A10, 60F05, 60H15.
Keywords: Airy1 process, association, ergodicity, CLT.

##  1 Introduction and main results

The Airy1 process, $\\{\mathcal{A}_{1}(x):x\in{\mathbb{R}}\\}$, derived by
Sasamoto [27] in the context of TASEP, is a stationary stochastic process
whose finite-dimensional distributions are given in terms of the Fredholm
determinant (see [31, Section 4.1.2]). In particular, the one-point
distribution is a scalar multiple of the GOE Tracy-Widom distribution, namely,

$\displaystyle\mathrm{P}\\{\mathcal{A}_{1}(0)\leq s\\}=F_{1}(2s),\quad
s\in{\mathbb{R}},$

where $F_{1}$ denotes the GOE Tracy-Widom distribution. Moreover, it was
proven by Quastel and Remenik [25] that the sample paths are Hölder continuous
with exponent $\frac{1}{2}-$.

Recently, Basu et al [4] have shown that the covariance of the Airy1 process
decays super-exponentially by showing that $\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(0))=\mathrm{e}^{-(\frac{4}{3}+o(1))x^{3}}$
as $x\to\infty$. This immediately implies the law of large numbers, i.e., as
$N\to\infty$,

$\displaystyle\frac{1}{N}\int_{0}^{N}\mathcal{A}_{1}(x)\
\mathrm{d}x\to\mathrm{E}[\mathcal{A}_{1}(0)],\quad\text{ in $L^{2}(\Omega)$}.$

We are aiming to establish the ergodicity and the matching central limit
theorem for the Airy1 process.

######  Theorem 1.1.

$\\{\mathcal{A}_{1}(x):x\in{\mathbb{R}}\\}$ is ergodic. And as $N\to\infty$,

$\displaystyle\frac{1}{\sqrt{N}}\int_{0}^{N}(\mathcal{A}_{1}(x)-\mathrm{E}[\mathcal{A}_{1}(x)])\
\mathrm{d}x\xrightarrow{\rm d}{\rm N}(0,\sigma^{2}),$ (1.1)

where $\sigma^{2}=\int_{\mathbb{R}}\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(0))\mathrm{d}x\in(0,\infty)$, and
“$\xrightarrow{\text{\rm d}\,}$” denotes convergence in distribution.

Note that Quastel and Remenik [25] proved that the Airy1 process fluctuates
locally like a Brownian motion (see [25, Theorem 3]), using a formula for the
$n$-dimensional distributions of the Airy1 process in terms of a Fredholm
determinant on $L^{2}({\mathbb{R}})$. We refer to Hägg [17] for the local
central limit theorem for the Airy2 process.

The main tool to prove the ergodicity property and CLT (1.1) in Theorem 1.1 is
the association property of of the Airy1 process (see Section 2 for the
definition of association of stochastic process and related information).
Indeed, we have the following.

######  Theorem 1.2.

$\\{\mathcal{A}_{1}(x):x\in{\mathbb{R}}\\}$ is associated.

The association property of the Airy1 process leads to a fundamental
inequality, known as Newman’s inequality (see (2.2) below), which plays a
crucial role in proving ergodicity and CLT for the Airy1 process. Furthermore,
Newman’s inequality implies an inequality for characteristic functions (see
(2.4) below), which together with the asymptotic behavior of the GOE Tracy-
Widom distribution, enables us to derive a Poisson limit theorem for high-
level exceedances of the Airy1 process.

Denote by $\rm T$ the inverse function of $s\mapsto 1-F_{1}(s)$. We have the
following Poisson approximation for the Airy1 process.

######  Theorem 1.3.

Fix $\lambda>0$. Let $\\{x_{k,N}\\}_{N\geq 1,1\leq k\leq
N}\subset{\mathbb{R}}$ satisfy

$\displaystyle\liminf_{N\to\infty}\frac{\min_{1\leq j\neq k\leq
N}|x_{j,N}-x_{k,N}|}{(3\log N)^{1/3}}>1.$ (1.2)

Then, as $N\to\infty$,

$\displaystyle\sum_{k=1}^{N}\bm{1}_{\\{\mathcal{A}_{1}(x_{k,N})>\frac{1}{2}{\rm
T}(\lambda/N)\\}}\xrightarrow{\rm d}{\rm Poisson}(\lambda),$ (1.3)

where ${\rm Poisson}(\lambda)$ denotes a Poisson random variable with
parameter $\lambda$.

Conus et al [12] studied the spatial asymptotic behavior for the solution to
stochastic heat equation using a localization argument (see Chen [11] for
precise spatial asymptotic behavior for the parabolic Anderson equation). This
localization argument is based on the mild formula for the stochastic heat
equation and does not apply to the Airy1 process. However, the association
property provides a way to analyze the independence structure of the Airy1
process in terms of the asymptotic behavior of the covariance (see the
inequality (6) below). Combined with the asymptotic behavior of the tail
probability of the maximum of the Airy1 process (see Proposition 6.1 below),
we can study the precise asymptotic of the maximum of the Airy1 process.

######  Theorem 1.4.

Almost surely,

$\displaystyle\lim_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)}{(\log
N)^{2/3}}=\frac{1}{2}\left(\frac{3}{2}\right)^{2/3}.$ (1.4)

As for the Airy2 process $\\{\mathcal{A}_{2}(x):x\in{\mathbb{R}}\\}$, we have
the following.

######  Theorem 1.5.

Almost surely,

$\displaystyle\left(\frac{1}{4}\right)^{2/3}\leq\liminf_{N\to\infty}\frac{\max_{0\leq
x\leq N}\mathcal{A}_{2}(x)}{(\log
N)^{2/3}}\leq\limsup_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{2}(x)}{(\log N)^{2/3}}\leq\left(\frac{3}{4}\right)^{2/3}.$
(1.5)

The paper is organized as following. After introducing some preliminaries in
Section 2, we first prove Theorem 1.2 in Section 3. Then we establish the
ergodicity and CLT in Section 4. The Poisson approximation (Theorem 1.3) is
established in Section 5. Finally, we prove Theorems 1.4 and 1.5 in Section 6.

Let us close the introduction with a brief description of the notation of this
paper. For a Lipschitz continuous function
$g:{\mathbb{R}}^{m}\mapsto{\mathbb{R}}$, define

$\text{\rm Lip}(g):=\sup_{a,b\in{\mathbb{R}}^{m},a\neq
b}\frac{|g(b)-g(a)|}{\|b-a\|}$

where $\|\cdot\|$ denotes the Euclidean norm on ${\mathbb{R}}^{m}$, and for
$j=1,\ldots,m$, we define

$\displaystyle\text{\rm Lip}_{j}(g):=\sup_{a_{j}\neq
b_{j}}\frac{|g(a)-g(b)|}{|a_{j}-b_{j}|},$

where $a_{j}$ and $b_{j}$ are the $j$-th coordinates of $a$ and $b$
respectively.

##  2 Preliminaries

In this section, we will first recall some basic facts on associated random
variables. We refer to [24] for more details on associated sequences.

First, let us recall the definition of association; see Esary, Proschan, and
Walkup [16]. A random vector $X:=(X_{1}\,,\ldots,X_{m})$ is said to be
_associated_ if

$\operatorname{\text{\rm Cov}}[h_{1}(X)\,,h_{2}(X)]\geq 0,$ (2.1)

for every pair of functions $h_{1},h_{2}:{\mathbb{R}}^{m}\to{\mathbb{R}}$ that
are nondecreasing in every coordinate and satisfy $h_{1}(X),h_{2}(X)\in
L^{2}(\Omega)$. A random field $\Phi=\\{\Phi(x)\\}_{x\in{\mathbb{R}}^{d}}$ is
_associated_ if $(\Phi(x_{1})\,,\ldots,\Phi(x_{m}))$ is associated for every
$x_{1},\ldots,x_{m}\in{\mathbb{R}}^{d}$. We remark that an associated random
vector is also called to satisfy the FKG inequalities; see Newman [21].

Esary, Proschan, and Walkup [16] show that the random vector $X$ is associated
iff (2.1) holds for all bounded and continuous functions $h_{1},h_{2}$ that
are nondecreasing coordinatewise. Using approximation and dominated
convergence theorem, in order to verify that $X$ is associated, it suffices to
check that (2.1) holds for all $h_{1},h_{2}$ which are smooth, nondecreasing
coordinatewise and have bounded partial derivatives.

For an associated random vector $(X,Y)$, it is clear that the random variables
$X$ and $Y$ are positive quadrant dependent (see [19] for the definition of
positive quadrant dependence). Therefore, according to Newman [21, Lemma 3],
for an associated random vector $(X,Y)$ such that both $X$ and $Y$ have finite
variance, we have

$\displaystyle\left|\operatorname{\text{\rm
Cov}}(f(X)\,,g(Y))\right|\leq\|f^{\prime}\|_{\infty}\|g^{\prime}\|_{\infty}\operatorname{\text{\rm
Cov}}(X\,,Y),$ (2.2)

for $C^{1}$ complex valued functions $f$, $g$ on ${\mathbb{R}}$ with
$f^{\prime}$, $g^{\prime}$ bounded, where $\|\cdot\|_{\infty}$ denotes the sup
norm on ${\mathbb{R}}$. Inequality (2.2) is known as Newman’s inequality.
Bulinski [6] generalized (2.2) to Lipschitz continuous functions (see also
[24, Theorem 6.2.4]). Moreover, according to Bulinski and Shabanovich [7] (see
also [24, Theorem 6.2.6]), letting $(Y_{1},\ldots,Y_{m})$ be an associated
random vector such that each component has finite variance, then we have for
Lipschitz continuous functions $f,g:{\mathbb{R}}^{m}\mapsto{\mathbb{R}}$,

$\displaystyle\left|\operatorname{\text{\rm
Cov}}(f(Y_{1},\ldots,Y_{m})\,,g(Y_{1},\ldots,Y_{m}))\right|\leq\sum_{j=1}^{m}\sum_{\ell=1}^{m}\text{\rm
Lip}_{j}(f)\text{\rm Lip}_{\ell}(g)\operatorname{\text{\rm
Cov}}(Y_{j}\,,Y_{\ell}),$ (2.3)

where $\text{\rm Lip}_{j}(f),\text{\rm Lip}_{\ell}(g)$ are defined at the end
of Section 1. Inequality (2.3) generalizes [21, (12)], which is a multivariate
version of (2.2).

Another consequence of Newman’s inequality (2.2) is the following inequality
for the characteristic functions of associated random vector. Indeed, by [21,
Theorem 1], for an associated random vector $(Y_{1},\ldots,Y_{m})$ such that
each component has finite variance, we have for any real numbers
$r_{1},\ldots,r_{m}$,

$\displaystyle\left|\mathrm{E}\left[\mathrm{e}^{{\rm
i}\sum_{j=1}^{m}r_{j}Y_{j}}\right]-\prod_{j=1}^{m}\mathrm{E}\left[\mathrm{e}^{{\rm
i}r_{j}Y_{j}}\right]\right|\leq\frac{1}{2}\sum_{\begin{subarray}{c}1\leq
j,\ell\leq m\\\
j\neq\ell\end{subarray}}|r_{j}||r_{\ell}|\operatorname{\text{\rm
Cov}}(Y_{j}\,,Y_{\ell}).$ (2.4)

The study of central limit theorem for associated random sequences essentially
relies on the above inequality; see [21, Theorem 2]. We will see in Section 5
that inequality (2.4) also plays a crucial role in the study of Poisson
approximation for the Airy1 process. In order to prove Theorems 1.4 and 1.5,
we need the following probability inequalities for associated random vector.

######  Lemma 2.1.

Let $(Y_{1},\ldots,Y_{m})$ be an associated random vector. Let
$y_{1},\ldots,y_{m}$ be real numbers. For subsets $A,B$ of $\\{1,\ldots,m\\}$,

$\displaystyle\mathrm{P}\\{Y_{j}\leq y_{j},j\in A\cup
B\\}-\mathrm{P}\\{Y_{j}\leq y_{j},j\in A\\}\mathrm{P}\\{Y_{k}\leq y_{k},j\in
B\\}$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\leq\sum_{j\in
A}\sum_{k\in B}\left(\mathrm{P}\\{Y_{j}\leq y_{j},Y_{k}\leq
y_{k}\\}-\mathrm{P}\\{Y_{j}\leq y_{j}\\}\mathrm{P}\\{Y_{k}\leq
y_{k}\\}\right).$ (2.5)

As a consequence,

$\displaystyle\mathrm{P}\\{Y_{j}\leq y_{j},1\leq j\leq
m\\}-\prod_{j=1}^{m}\mathrm{P}\\{Y_{j}\leq y_{j}\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\leq\sum_{1\leq j<k\leq
m}\left(\mathrm{P}\\{Y_{j}\leq y_{j},Y_{k}\leq y_{k}\\}-\mathrm{P}\\{Y_{j}\leq
y_{j}\\}\mathrm{P}\\{Y_{k}\leq y_{k}\\}\right).$ (2.6)

###### Proof.

The proof of (2.1) is similar to that of [24, Theorem 1.2.2] (see also [18,
Lemma 1]). We define the random variables $Z_{j}=\bm{1}_{\\{Y_{j}\leq
y_{j}\\}}$, for $j=1,\ldots,m$. It is clear that $(Z_{1},\ldots,Z_{m})$ is
associated. Then we can follow along the same lines as in the proof of [24,
Theorem 1.2.2] to conclude (2.1).

Clearly, the inequality (2.1) holds with $m=2$. Assume that the inequality
(2.1) is true for $m-1$. Then for integer $m$,

$\displaystyle\mathrm{P}\\{Y_{j}\leq y_{j},1\leq j\leq
m\\}-\prod_{j=1}^{m}\mathrm{P}\\{Y_{j}\leq y_{j}\\}$
$\displaystyle\qquad\qquad=\mathrm{P}\\{Y_{j}\leq y_{j},1\leq j\leq
m\\}-\mathrm{P}\\{Y_{j}\leq y_{j},1\leq j\leq m-1\\}\mathrm{P}\\{Y_{m}\leq
y_{m}\\}$ $\displaystyle\qquad\qquad\quad+\mathrm{P}\\{Y_{j}\leq y_{j},1\leq
j\leq m-1\\}\mathrm{P}\\{Y_{m}\leq
y_{m}\\}-\prod_{j=1}^{m}\mathrm{P}\\{Y_{j}\leq y_{j}\\}$
$\displaystyle\qquad\qquad\leq\sum_{j=1}^{m-1}\left(\mathrm{P}\\{Y_{i}\leq
y_{j},Y_{m}\leq y_{m}\\}-\mathrm{P}\\{Y_{j}\leq y_{j}\\}\mathrm{P}\\{Y_{m}\leq
y_{m}\\}\right)$ $\displaystyle\qquad\qquad\quad+\sum_{1\leq j<k\leq
m-1}\left(\mathrm{P}\\{Y_{j}\leq y_{j},Y_{k}\leq
y_{k}\\}-\mathrm{P}\\{Y_{j}\leq y_{j}\\}\mathrm{P}\\{Y_{k}\leq
y_{k}\\}\right)$ $\displaystyle\qquad\qquad=\sum_{1\leq j<k\leq
m}\left(\mathrm{P}\\{Y_{j}\leq y_{j},Y_{k}\leq y_{k}\\}-\mathrm{P}\\{Y_{j}\leq
y_{j}\\}\mathrm{P}\\{Y_{k}\leq y_{k}\\}\right),$

where in the inequality, we have used (2.1) and the assumption for $m-1$.
Hence, we prove the inequality (2.1) by induction. ∎

We next introduce some facts on the GOE and GUE Tracy-Widom distributions,
denoted by $F_{1}$ and $F_{2}$ respectively. Let $F_{1}^{\prime}$ and
$F_{2}^{\prime}$ the derivatives of $F_{1}$ and $F_{2}$ respectively, which
correspond to the probability density functions of the GOE and GUE Tracy-Widom
distributions. Recall from [1] (see also [28, 29]) that $F_{1}$ and $F_{2}$
can be written as

$\displaystyle F_{1}(s)=F(s)E(s),\qquad F_{2}(s)=F(s)^{2},$ (2.7)

where

$\displaystyle
F(s)=\exp\left(-\frac{1}{2}\int_{s}^{\infty}R(r)\mathrm{d}r\right),\quad
E(s)=\exp\left(-\frac{1}{2}\int_{s}^{\infty}q(r)\mathrm{d}r\right).$ (2.8)

Here the (real) function $q$ is the solution to the Painlevé II equation

$\displaystyle q^{\prime\prime}=2q^{3}+sq,$

that satisfies the boundary condition

$\displaystyle
q(s)\sim\frac{1}{2\sqrt{\pi}s^{1/4}}\mathrm{e}^{-\frac{2}{3}s^{3/2}},\quad
s\to+\infty.$ (2.9)

The function $R$ is defined as

$\displaystyle R(s)=\int_{s}^{\infty}(q(r))^{2}\mathrm{d}r.$

Taking derivative in (2.7) and using the formulas in (2.8), we have

$\displaystyle F_{1}^{\prime}(s)=\frac{1}{2}F_{1}(s)\left(R(s)+q(s)\right).$
(2.10)

According to the asymptotics of the functions $q,R$ and $F_{1}$ as
$s\to-\infty$ (see [1, (11), (12), (18)]), we derive from (2.10) that the
probability density function $F^{\prime}_{1}$ is continuous and bounded.
Similarly, we have

$\displaystyle F_{2}^{\prime}(s)=F_{2}(s)R(s),$

and use the asymptotics of the functions $R$ and $F_{2}$ as $s\to-\infty$ (see
[1, (12), (16)]), we deduce that the GUE Tracy-Widom distribution also has a
bounded and continuous probability density function.

Moreover, in light of (2.9), we deduce that for all sufficiently large $s$,

$\displaystyle
R(s)\leq\int_{s}^{\infty}\frac{1}{\sqrt{\pi}r^{1/4}}\mathrm{e}^{-\frac{2}{3}r^{3/2}}q(r)\mathrm{d}r\leq\mathrm{e}^{-\frac{2}{3}s^{3/2}}.$

Hence, we combine with (2.10) and (2.9) to obtain that there exists a constant
$C_{1}>0$ such that

$\displaystyle
F_{1}^{\prime}(s)\leq\mathrm{e}^{-\frac{2}{3}s^{3/2}},\quad\text{for all
$s\geq C_{1}$}.$ (2.11)

Furthermore, we see from [14, Theorem 1] that for fixed
$\epsilon\in(0,\frac{2}{3})$, there exists a positive constant $C_{2}$
depending on $\epsilon$ such that

$\displaystyle\mathrm{e}^{-(\frac{2}{3}+\epsilon)s^{3/2}}\leq
1-F_{1}(s)\leq\mathrm{e}^{-(\frac{2}{3}-\epsilon)s^{3/2}},\quad\text{for all
$s\geq C_{2}$}.$ (2.12)

Recall that $\rm T$ denotes the inverse function of $s\mapsto 1-F_{1}(s)$. The
following lemma will be used when we prove Theorem 1.3 in Section 5.

######  Lemma 2.2.

Fix $\lambda>0$ and $\epsilon\in(0,\frac{2}{3})$. There exist an integer
$N_{0}$ and a positive constant $C$ both depending on $\lambda$ and $\epsilon$
such that for all $z\geq N_{0}$

$\displaystyle{\rm T}(\lambda/z)-{\rm T}(\lambda/(z-1))\geq
Cz^{-\frac{2+6\epsilon}{2+3\epsilon}}.$ (2.13)

###### Proof.

Taking derivative on both sides of the following identity

$\displaystyle\int_{{\rm
T}(\lambda/z)}^{\infty}F^{\prime}_{1}(s)\mathrm{d}s=\frac{\lambda}{z},\quad
z>\lambda,$

we observe that for all $z>\lambda$

$\displaystyle\frac{\rm d}{\mathrm{d}z}{\rm
T}(\lambda/z)=\frac{\lambda}{z^{2}F_{1}^{\prime}({\rm T}(\lambda/z))}.$ (2.14)

We can choose a sufficiently large integer $N_{0}$ depending on $\lambda>0$
and $\epsilon\in(0,\frac{2}{3})$ such that

$\displaystyle{\rm T}(\lambda/z)>C_{1}\vee C_{2},\quad\text{for all $z\geq
N_{0}-1$},$

where $C_{1},C_{2}$ are the constants in (2.11) and (2.12) respectively.
Hence, by (2.14) and (2.11), we derive that for all $z\geq N_{0}-1$,

$\displaystyle\frac{\rm d}{\mathrm{d}z}{\rm
T}(\lambda/z)\geq\frac{\lambda}{z^{2}}\mathrm{e}^{\frac{2}{3}([{\rm
T}(\lambda/z)]^{3/2})}\geq\frac{\lambda}{z^{2}}\left(\frac{z}{\lambda}\right)^{\frac{\frac{2}{3}}{\frac{2}{3}+\epsilon}}=Cz^{-\frac{2+6\epsilon}{2+3\epsilon}},$

where the second inequality holds by the first inequality in (2.12). Finally,
the above estimate and the mean-value theorem imply (2.13). ∎

##  3 Association

We prove the association property of the Airy1 process in this section. We
first consider the following stochastic heat equation

$\displaystyle\partial_{t}Z(t\,,x)=\delta\partial_{x}^{2}Z(t\,,x)+\frac{1}{4}\delta^{1/2}Z(t\,,x)\xi(t\,,x),\quad
t>0,x\in{\mathbb{R}}$ (3.1)

subject to flat initial date $Z(0)\equiv 1$ or narrow wedge initial data
$Z(0)=\delta_{0}$. In the above equation, $\delta$ is a positive constant and
$\xi$ denotes space-time white noise. Chen et al [10, Theorem A.4] have
established the association property for stochastic heat equation with general
diffusion coefficient and initial data, by showing that the Malliavin
derivative of the solution is non-negative almost surely (see [10, (A.4)]) and
then applying the Clark-Ocone formula and Ito’s isometry to show that the
covariance

$\displaystyle\operatorname{\text{\rm
Cov}}(h_{1}(Z(t_{1}\,,x_{1}),\ldots,Z(t_{m}\,,x_{m}))\,,h_{2}(Z(t_{1}\,,x_{1}),\ldots,Z(t_{m}\,,x_{m})))$

is nonnegative for all $h_{1},h_{2}$ which are smooth, nondecreasing
coordinatewise and have bounded partial derivatives. Although the constants
(depending on $\delta$) in equation (3.1) are different from those in equation
(1.1) of [10], Theorem A.4 of [10] ensures that the solution to (3.1) is
associated.

######  Theorem 3.1 ([10, Theorem A.4]).

For every fixed $\delta>0$, the solution
$\\{Z(t\,,x):(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}\\}$ to (3.1) with flat
or narrow wedge initial condition is associated.

Now we prove Theorem 1.2.

###### Proof of Theorem 1.2.

For $\delta>0$, define

$\displaystyle\tilde{h}_{\delta}(t\,,x)=4\delta\log
Z(t,x)+\frac{t}{12},\quad(t,x)\in(0,\infty)\times{\mathbb{R}},$ (3.2)

where $Z(t\,,x)$ is the solution to (3.1) subject to flat initial condition
$Z(0)\equiv 1$. It is clear that for each $\delta>0$, the process
$\tilde{h}_{\delta}=\\{\tilde{h}_{\delta}(t,x):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$
is associated. Let
$\mathfrak{h}=\\{\mathfrak{h}(t,x;0):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$
be the KPZ fixed point (see [20]) with flat initial data. According to [26,
Theorem 2.2(3)], as $\delta\to 0$, the finite-dimensional distributions of
$\tilde{h}_{\delta}$ converge to those of $\mathfrak{h}$; see also [30,
Theorem 1]. Since convergence in distribution preserves the association
property (see (P5) of [16]), we see that $\mathfrak{h}$ is associated.
Moreover, because $\\{\mathfrak{h}(1,x;0):x\in{\mathbb{R}}\\}$ has the same
finite-dimensional distributions as
$\\{2^{1/3}\mathcal{A}_{1}(2^{-2/3}x):x\in{\mathbb{R}}\\}$ (see [20, (4.15)]),
we conclude that the Airy1 process is associated. ∎

######  Proposition 3.2.

The Airy2 process $\\{\mathcal{A}_{2}(x):x\in{\mathbb{R}}\\}$ is associated.

###### Proof.

The proof is similar to that of Theorem 1.2. Denote by
$\\{\mathfrak{h}(t,x;\partial_{0}):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$
the KPZ fixed point starting from narrow wedge at $0$. Let
$\tilde{h}_{\delta}(t\,,x)$ be defined as in (3.2), where $Z(t\,,x)$ is the
solution to (3.1) subject to narrow wedge initial condition
$Z(0)\equiv\delta_{0}$. By Theorem 3.1, we know that for each $\delta>0$, the
process $\\{\tilde{h}_{\delta}(t,x):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$
is associated. Theorem 2.2(3) of [26] ensures that the finite-dimensional
distributions of
$\\{\tilde{h}_{\delta}(t,x):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$ converge
to those of
$\\{\mathfrak{h}(t,x;\partial_{0}):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$ as
$\delta\to 0$. This implies that the process
$\\{\mathfrak{h}(t,x;\partial_{0}):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$ is
associated. In particular, the process
$\\{\mathfrak{h}(1,x;\partial_{0}):x\in{\mathbb{R}}\\}$ is associated and
hence the process
$\\{\mathfrak{h}(1,x;\partial_{0})+x^{2}:x\in{\mathbb{R}}\\}$ is also
associated. Moreover, since the process
$\\{\mathfrak{h}(1,x;\partial_{0})+x^{2}:x\in{\mathbb{R}}\\}$ has the same
finite-dimensional distributions as the Airy2 process
$\\{\mathcal{A}_{2}(x):x\in{\mathbb{R}}\\}$ (see [20, (4.14)]), we conclude
that the Airy2 process is associated. ∎

##  4 Ergodicity and CLT

We prove the ergodicity and central limit theorem for the Airy1 process in
this section. We first prove the ergodicity.

###### Proof of Theorem 1.1: ergodicity.

Chen et al [9, Lemma 7.2] present a general criterion on the ergodicity for
stationary process, which has been modified to a variant by Balan and Zheng
[3]. According to Balan and Zheng [3, Lemma 4.2], in order to prove that the
Airy1 process is ergodic, we need to verify that

$\displaystyle\lim_{N\to\infty}\frac{1}{N^{2}}\text{\rm
Var}\left(\int_{[0,N]}G\left(\sum_{j=1}^{k}b_{i}\mathcal{A}_{1}(x+\zeta_{j})\right)\mathrm{d}x\right)=0,$
(4.1)

for all inetegers $k\geq 1$, for every $b_{1},\ldots,b_{k}$,
$\zeta_{1},\ldots,\zeta_{k}\in{\mathbb{R}}$, and for $G\in\\{x\mapsto\cos
x,x\mapsto\sin x\\}$.

Notice that

$\displaystyle\frac{1}{N^{2}}\text{\rm
Var}\left(\int_{[0,N]}G\left(\sum_{j=1}^{k}b_{i}\mathcal{A}_{1}(x+\zeta_{j})\right)\mathrm{d}x\right)$
$\displaystyle\qquad\qquad=\frac{1}{N^{2}}\int_{[0,N]^{2}}\mathrm{d}x\mathrm{d}y\,\operatorname{\text{\rm
Cov}}\left(G\left(\sum_{j=1}^{k}b_{j}\mathcal{A}_{1}(x+\zeta_{j})\right)\,,G\left(\sum_{j=1}^{k}b_{j}\mathcal{A}_{1}(y+\zeta_{j})\right)\right).$
(4.2)

Since the Airy1 process is associated, we apply the inequality (2.3) with
$Y_{j}=\mathcal{A}_{1}(x+\zeta_{j})$ for $j=1,\ldots,k$,
$Y_{j}=\mathcal{A}_{1}(y+\zeta_{j-k})$ for $j=k+1,\ldots,2k$,
$f(y_{1},\ldots,y_{2k})=G(\sum_{j=1}^{k}b_{j}y_{j})$, and
$g(y_{1},\ldots,y_{2k})=G(\sum_{j=k+1}^{2k}b_{j-k}y_{j})$, to see that

$\displaystyle\operatorname{\text{\rm
Cov}}\left(G\left(\sum_{j=1}^{k}b_{i}\mathcal{A}_{1}(x+\zeta_{j})\right)\,,G\left(\sum_{j=1}^{k}b_{i}\mathcal{A}_{1}(y+\zeta_{j})\right)\right)$
$\displaystyle\qquad\qquad\qquad\qquad\leq\sum_{j=1}^{k}\sum_{\ell=1}^{k}|b_{j}||b_{\ell}|\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x+\zeta_{j})\,,\mathcal{A}_{1}(y+\zeta_{\ell})).$

Here, we also use the fact that $\text{\rm Lip}_{j}(f)\leq|b_{j}|$ for $1\leq
j\leq k$, $\text{\rm Lip}_{j}(f)=0$ for $j=k+1,\ldots,2k$ and $\text{\rm
Lip}_{\ell}(g)=0$ for $1\leq\ell\leq k$, $\text{\rm
Lip}_{\ell}(g)\leq|b_{\ell-k}|$ for $\ell=k+1,\ldots,2k$. According to [4,
Corollary 3.2], there exist constants $c_{1},c_{2}>0$ such that for all
$x\in{\mathbb{R}}$,

$\displaystyle\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(0))\leq
K(x):=c_{1}\mathrm{e}^{-c_{2}x^{2}}.$

Hence, the left-hand side of (4) is bounded above by

$\displaystyle\frac{\sup_{1\leq j\leq
k}b_{j}^{2}}{N^{2}}\sum_{j=1}^{k}\sum_{\ell=1}^{k}\int_{[0,N]^{2}}K(x-y+\zeta_{j}-\zeta_{\ell})\mathrm{d}x\mathrm{d}y$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\left[\sup_{1\leq
j\leq
k}b_{j}^{2}\right]\sum_{j=1}^{k}\sum_{\ell=1}^{k}\left(\bm{1}_{N}*\tilde{\bm{1}}_{N}*K\right)(\zeta_{j}-\zeta_{\ell}),$

where the function $\bm{1}_{N}$ is defined as
$\bm{1}_{N}(x)=\frac{1}{N}\bm{1}_{[0,N]}(x)$ for $x\in{\mathbb{R}}$ and
$\tilde{\bm{1}}_{N}(x)=\bm{1}_{N}(-x)$. The function
$\bm{1}_{N}*\tilde{\bm{1}}_{N}*K$ is non-negative definite and hence maximized
at origin. Hence we have

$\displaystyle\frac{1}{N^{2}}\text{\rm
Var}\left(\int_{[0,N]}g\left(\sum_{j=1}^{k}b_{i}\mathcal{A}_{1}(x+\zeta_{j})\right)\mathrm{d}x\right)$
$\displaystyle\leq\sup_{1\leq j\leq
k}b_{j}^{2}\,k^{2}\left(\bm{1}_{N}*\tilde{\bm{1}}_{N}*K\right)(0)$
$\displaystyle\leq\sup_{1\leq j\leq
k}b_{j}^{2}\frac{k^{2}}{N}\int_{-N}^{N}K(x)\mathrm{d}x,$

which converges to $0$ as $N\to\infty$. Therefore, (4.1) is verified and we
complete the proof of ergodicity. ∎

######  Remark 4.1.

We refer to [13, 23] for the ergodicity of the Airy2 process.

We proceed to prove the central limit theorem (1.1).

###### Proof of Theorem 1.1: CLT.

We will apply [21, Theorem 2] (see also [24, Theorem 1.2.19]) to prove the
central limit theorem in Theorem 1.1. We can assume that the limit in (1.1) is
taken along integers. Denote

$\displaystyle X_{k}=\int_{k}^{k+1}\mathcal{A}_{1}(x)\mathrm{d}x,\quad
k\in\mathbb{Z}.$

Since the Airy1 process is stationary and $\mathcal{A}_{1}(0)$ has finite
second moment, the process $\\{X_{k}:k\in\mathbb{Z}\\}$ is stationary and
$X_{0}$ has finite second moment. For each $k\in\mathbb{Z}$, let

$\displaystyle
X_{k}^{(n)}=\frac{1}{n}\sum_{j=1}^{n}\mathcal{A}_{1}(k+\frac{j}{n}),\quad
n\geq 1.$

Clearly, for each $n\geq 1$, the process $\\{X_{k}^{(n)}:k\in\mathbb{Z}\\}$ is
associated. Moreover, since the Airy1 process has continuous sample paths (see
[25]), we know that as $n\to\infty$, the finite-dimensional distributions of
$\\{X_{k}^{(n)}:k\in\mathbb{Z}\\}$ converge to those of
$\\{X_{k}:k\in\mathbb{Z}\\}$. Hence, the process $\\{X_{k}:k\in\mathbb{Z}\\}$
is associated.

In order to apply [21, Theorem 2], it remains to verify that

$\displaystyle\sigma^{2}:=\sum_{k\in\mathbb{Z}}\operatorname{\text{\rm
Cov}}(X_{0}\,,X_{k})$

is finite. Indeed, by the stationarity of the Airy1 process,

$\displaystyle\sigma^{2}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\operatorname{\text{\rm
Cov}}\left(\int_{0}^{1}\mathcal{A}_{1}(x)\mathrm{d}x\,,\int_{k}^{k+1}\mathcal{A}_{1}(y)\mathrm{d}y\right)$
$\displaystyle=\sum_{k\in\mathbb{Z}}\int_{0}^{1}\mathrm{d}x\int_{k}^{k+1}\mathrm{d}y\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(y))$
$\displaystyle=\int_{0}^{1}\mathrm{d}x\int_{\mathbb{R}}\mathrm{d}y\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x-y)\,,\mathcal{A}_{1}(0))=\int_{\mathbb{R}}\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(0))\mathrm{d}x,$

which is finite by [4, Corollary 3.2].

Therefore, all the conditions in [21, Theorem 2] are met and we conclude that
as $N\to\infty$,

$\displaystyle\frac{X_{0}+\ldots+X_{N-1}-N\mathrm{E}[X_{0}]}{\sqrt{N}}\xrightarrow{\rm
d}{\rm N}(0,\sigma^{2}),$

which implies (1.1). ∎

######  Remark 4.2.

(1). We can also apply [21, Theorem 2] to derive a discrete version of (1.1),
that is, as $N\to\infty$,

$\displaystyle\frac{\mathcal{A}_{1}(1)+\ldots+\mathcal{A}_{1}(N)-N\mathrm{E}[\mathcal{A}_{1}(0)]}{\sqrt{N}}\xrightarrow{\rm
d}{\rm N}(0,\tau^{2})$

with $\tau^{2}=\sum_{k\in\mathbb{Z}}\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(k)\,,\mathcal{A}_{1}(0))$.

(2). One can apply the Berry-Esseen theorem for associated sequences (see for
instance [24, p.15-16]) to obtain the rate of convergence in the central limit
theorem for the Airy1 process.

(3). Recall that
$\\{\mathfrak{h}(t,x;0):(t,x)\in(0,\infty)\times{\mathbb{R}}\\}$ denotes the
KPZ fixed point (see [20]) with flat initial data. Using the scaling
invariance property of the KPZ fixed point (see [20, Theorem 4.5(i)]) and
(1.1), we deduce that for fixed $t>0$, as $N\to\infty$,

$\displaystyle\frac{1}{\sqrt{N}}\int_{0}^{N}\left(\mathfrak{h}(t,x;0)-\mathrm{E}[\mathfrak{h}(t,x;0)]\right)\,\mathrm{d}x\xrightarrow{\rm
d}{\rm N}(0,2^{4/3}t^{4/3}\sigma^{2}),$

where $\sigma^{2}$ is the quantity given in Theorem 1.1. One may expect that a
corresponding functional central limit theorem in $t$ also holds.

Similarly, we can also establish the central limit theorem for the Airy2
process.

######  Proposition 4.3.

As $N\to\infty$,

$\displaystyle\frac{1}{\sqrt{N}}\int_{0}^{N}(\mathcal{A}_{2}(x)-\mathrm{E}[\mathcal{A}_{2}(x)])\
\mathrm{d}x\xrightarrow{\rm d}{\rm N}(0,\tilde{\sigma}^{2}),$

where $\tilde{\sigma}^{2}=\int_{\mathbb{R}}\operatorname{\text{\rm
Cov}}(\mathcal{A}_{2}(x)\,,\mathcal{A}_{2}(0))\mathrm{d}x\in(0,\infty)$.

###### Proof.

The proof follows along the same lines as that of (1.1), applying [21, Theorem
2] with the stationarity (see [23]), association (see Proposition 3.2) of the
Airy2 process and the fact that the covariance of the Airy2 process is
integrable since the decay rate of $\operatorname{\text{\rm
Cov}}(\mathcal{A}_{2}(x)\,,\mathcal{A}_{2}(0))$ is $x^{-2}$ as $x\to\infty$
(see [32]). ∎

We conclude this section by another application of Newman’s inequality, which
is used to study the Lebesgue measure of the set $x\in[0,N]$ where
$\mathcal{A}_{1}(x)$ exceeds a high level. The following result is analogous
to [9, Corollary 9.5], where the Poincaré inequality is used to estimate the
covariance of the solution to stochastic partial differential equation. In the
context of the Airy1 process, when it comes to the estimate of covariance,
Newman’s inequality plays the role as the Poincaré inequality.

######  Corollary 4.4.

For $\alpha\in(0,3/4)$, as $N\to\infty$,

$\displaystyle\frac{1}{\log
N}\log\left(\int_{0}^{N}\bm{1}_{\\{\mathcal{A}_{1}(x)>\frac{1}{2}(\alpha\log
N)^{2/3}\\}}\mathrm{d}x\right)\to 1-\frac{2}{3}\alpha,\quad\text{in
probability}.$ (4.3)

###### Proof.

Choose and fix $\alpha\in(0,3/4)$. For $N\geq 1$, we define
$a_{N}=\frac{1}{2}(\alpha\log N)^{2/3}$ and two functions

$\displaystyle G_{N}(z):=1\wedge(z-a_{N}+1)_{+}\quad\text{and}\quad
g_{N}=1\wedge(z-a_{N})_{+},\quad z\in{\mathbb{R}}.$

Recall that $\mathrm{P}\\{\mathcal{A}_{1}(0)\leq s\\}=F_{1}(2s)$, where
$F_{1}$ denotes the GOE Tracy-Widom distribution function. According to [14,
Theorem 1],
$s^{-3/2}\log\mathrm{P}\\{\mathcal{A}_{1}(0)>\frac{1}{2}s\\}\to-\frac{2}{3}$
as $s\to+\infty$. By stationarity of the Airy1 process, we see that as
$N\to\infty$,

$\displaystyle\mathrm{E}\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x=N^{1-\frac{2}{3}\alpha+o(1)}\quad\text{and}\quad\mathrm{E}\int_{0}^{N}G_{N}(\mathcal{A}_{1}(x))\mathrm{d}x=N^{1-\frac{2}{3}\alpha+o(1)}.$
(4.4)

Using Chebychev’s inequality, we have for any fixed $\epsilon\in(0,1)$,

$\displaystyle\mathrm{P}\left\\{\left|\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x-\mathrm{E}\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right|>\epsilon\,\mathrm{E}\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq\frac{\text{\rm
Var}\left(\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right)}{\epsilon^{2}\left|\mathrm{E}\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right|^{2}}.$
(4.5)

Notice that for all $N\geq 1$

$\displaystyle\text{\rm
Var}\left(\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right)$
$\displaystyle=\int_{[0,N]^{2}}\mathrm{d}x\mathrm{d}y\,\operatorname{\text{\rm
Cov}}(g_{N}(\mathcal{A}_{1}(x))\,,g_{N}(\mathcal{A}_{1}(y)))$
$\displaystyle\leq[\text{\rm
Lip}(g_{N})]^{2}\int_{[0,N]^{2}}\mathrm{d}x\mathrm{d}y\,\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(y))$
$\displaystyle=\int_{[0,N]^{2}}\mathrm{d}x\mathrm{d}y\,\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(y))\leq
N\int_{\mathbb{R}}\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(0))\mathrm{d}x,$ (4.6)

where the first inequality holds by the association property of the Airy1
process and (2.3), and the second equality is because of $\text{\rm
Lip}(g_{N})=1$. Hence, we substitute (4) into (4) to see that

$\displaystyle\mathrm{P}\left\\{\left|\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x-\mathrm{E}\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right|>\epsilon\,\mathrm{E}\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq\epsilon^{-2}N^{\frac{4}{3}\alpha-1+o(1)}\int_{\mathbb{R}}\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(0))\mathrm{d}x.$

Since $\alpha\in(0,3/4)$, we obtain that as $N\to\infty$,

$\displaystyle\frac{\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x}{\mathrm{E}\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x}\to
1\quad\text{in probability}.$

Taking logarithm and using (4.4), we conclude that as $N\to\infty$,

$\displaystyle\frac{\log\left(\int_{0}^{N}g_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right)}{\log
N}\to 1-\frac{2}{3}\alpha\quad\text{in probability}.$

Similarly, we can prove that as $N\to\infty$,

$\displaystyle\frac{\log\left(\int_{0}^{N}G_{N}(\mathcal{A}_{1}(x))\mathrm{d}x\right)}{\log
N}\to 1-\frac{2}{3}\alpha\quad\text{in probability}.$

Since $g_{N}\leq\bm{1}_{(a_{N},\infty)}\leq G_{N}$, the preceding displays
imply (4.3) and hence complete the proof. ∎

######  Remark 4.5.

The convergence in (4.3) also holds almost surely. This follows from a
standard subsequencing and sandwich type argument (see the proof of [15,
Theorem 2.3.9]). We omit the details here.

##  5 Poisson approximation

We prove Theorem 1.3 in this section. The proof of Theorem 1.3 is based on
inequality (2.4) and some standard arguments; see [22, Theorem 11] for a
general result on limit theorems for sums of associated variables.

The indicator function in (1.3) is not Lipschitz continuous. In order to apply
the inequalities (2.3) and (2.4) to derive the Poisson limit theorem, we will
first approximate the indicator function in (1.3) by Lipschitz continuous
function. Fix $\lambda>0$. For $N\geq 2$, we introduce the function

$\displaystyle g_{N}(z)=1\wedge\frac{(z-\frac{1}{2}{\rm
T}(\lambda/N))_{+}}{[{\rm T}(\lambda/(N+1))-{\rm T}(\lambda/N)]/2},\quad
z\in{\mathbb{R}},$ (5.1)

where we recall that $\rm T$ denotes by the inverse function of $s\mapsto
1-F_{1}(s)$. It is clear that for all $N\geq 2$,

$\displaystyle g_{N}(z)\leq\bm{1}_{(\frac{1}{2}{\rm
T}(\lambda/N),\infty)}(z)\leq g_{N-1}(z),\quad\text{for all
$z\in{\mathbb{R}}$},$ (5.2)

and

$\displaystyle\text{\rm Lip}(g_{N})=\frac{2}{{\rm T}(\lambda/(N+1))-{\rm
T}(\lambda/N)}.$ (5.3)

We first prove the following result on Poisson approximation for the Airy1
process.

######  Proposition 5.1.

Fix $\lambda>0$. Let $g_{N}$ be defined as in (5.1) for $N\geq 2$. Let
$\\{x_{k,N\\}_{N\geq 1,1\leq k\leq N}}\subset{\mathbb{R}}$ satisfy the
condition in (1.2). Then, as $N\to\infty$,

$\displaystyle\sum_{k=1}^{N}g_{N}(\mathcal{A}_{1}(x_{k,N}))\xrightarrow{\rm
d}{\rm Poisson}(\lambda).$ (5.4)

###### Proof.

Step 1. Let $U_{1,N},\ldots,U_{N,N}$ be independent random variables, each
having the same distribution as $g_{N}(\mathcal{A}_{1}(0))$ (for this, we can
enlarge the probability space if necessary). The first task will be to show
that

$\displaystyle\sum_{k=1}^{N}U_{k,N}\xrightarrow{\rm d}{\rm
Poisson}(\lambda),\quad\text{as $N\to\infty$}.$ (5.5)

Notice that the law of the family of the random variables
$\sum_{k=1}^{N}U_{k,N},N\geq 1$ is tight. This is because that by the
definition of $g_{N}$ in (5.1),

$\displaystyle\sup_{N\geq
1}\mathrm{E}\left[\sum_{k=1}^{N}U_{k,N}\right]=\sup_{N\geq
1}\left(N\mathrm{E}[g_{N}(\mathcal{A}_{1}(0))]\right)\leq\sup_{N\geq
1}\left(N\mathrm{E}\left[\bm{1}_{(\frac{1}{2}{\rm
T}(\lambda/N),\infty)}(\mathcal{A}_{1}(0))\right]\right)=\lambda,$

where the first inequality is due to the first inequality in (5.2). Thus, in
order to prove (5.5), it suffices to prove that the moment generating function
of $\sum_{k=1}^{N}U_{k,N}$ converges to that of ${\rm Poisson}(\lambda)$, as
$N\to\infty$.

For $\theta\in{\mathbb{R}}$, define

$\displaystyle\phi_{N}(\theta)$
$\displaystyle:=\mathrm{E}\left[\mathrm{e}^{-\theta
g_{N}(\mathcal{A}_{1}(0))}\right]\quad\text{and}\quad\Phi_{N}(\theta):=\mathrm{E}\left[\mathrm{e}^{-\theta\sum_{k=1}^{N}U_{k,N}}\right]=\left(\phi_{N}(\theta)\right)^{N}.$

Thanks to (5.2), for all $\theta>0$, we have

$\displaystyle\phi_{N}(\theta)\geq\mathrm{E}\left[\mathrm{e}^{-\theta\bm{1}_{(\frac{1}{2}{\rm
T}(\lambda/N),\infty)}(\mathcal{A}_{1}(0))}\right]=\frac{\lambda}{N}\mathrm{e}^{-\theta}+(1-\frac{\lambda}{N})=1-\frac{\lambda}{N}(1-\mathrm{e}^{-\theta}),$

and

$\displaystyle\phi_{N}(\theta)\leq\mathrm{E}\left[\mathrm{e}^{-\theta\bm{1}_{(\frac{1}{2}{\rm
T}(\lambda/(N+1)),\infty)}(\mathcal{A}_{1}(0))}\right]=1-\frac{\lambda}{N+1}(1-\mathrm{e}^{-\theta}).$

This leads to that for all $\theta>0$,

$\displaystyle\left(1-\frac{\lambda}{N}(1-\mathrm{e}^{-\theta})\right)^{N}\leq\Phi_{N}(\theta)\leq\left(1-\frac{\lambda}{N+1}(1-\mathrm{e}^{-\theta})\right)^{N}.$
(5.6)

Letting $N\to\infty$, we obtain that for all $\theta>0$,

$\displaystyle\lim_{N\to\infty}\Phi_{N}(\theta)=\mathrm{e}^{-\lambda(1-\mathrm{e}^{-\theta})}.$

The above display also holds when $\theta\leq 0$ by switching the first and
third terms in (5.6). This verifies (5.5).

Step 2. We proceed to compare the characteristic functions of
$\sum_{k=1}^{N}g_{N}(\mathcal{A}_{1}(x_{k,N}))$ and $\sum_{k=1}^{N}U_{k,N}$.
First, we have for all $\theta\in{\mathbb{R}}$,

$\displaystyle\left|\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]-\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}U_{k,N}}\right]\right|$
$\displaystyle\qquad\qquad\qquad=\left|\mathrm{E}\left[\mathrm{e}^{{\rm
i}\sum_{k=1}^{N}\theta
g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]-\prod_{k=1}^{N}\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]\right|.$ (5.7)

Since the Airy1 process is associated and $g_{N}$ is a nondecreasing function,
the random vector
$(g_{N}(\mathcal{A}_{1}(x_{1,N})),\ldots,g_{N}(\mathcal{A}_{1}(x_{N,N}))$ is
associated. Clearly, each component of this random vector has finite variance.
Hence, by (2.4), we have for all $\theta\in{\mathbb{R}}$,

$\displaystyle\left|\mathrm{E}\left[\mathrm{e}^{{\rm i}\sum_{k=1}^{N}\theta
g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]-\prod_{k=1}^{N}\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]\right|$
$\displaystyle\qquad\qquad\qquad\leq\frac{1}{2}\sum_{\begin{subarray}{c}1\leq
j,k\leq N\\\ j\neq k\end{subarray}}\theta^{2}\operatorname{\text{\rm
Cov}}(g_{N}(\mathcal{A}_{1}(x_{j,N}))\,,g_{N}(\mathcal{A}_{1}(x_{k,N})))$
$\displaystyle\qquad\qquad\qquad\leq\frac{1}{2}[\text{\rm
Lip}(g_{N})]^{2}\theta^{2}\sum_{\begin{subarray}{c}1\leq j,k\leq N\\\ j\neq
k\end{subarray}}\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x_{j,N})\,,\mathcal{A}_{1}(x_{k,N})),$ (5.8)

where the second inequality is due to (2.3). According to [4, Theorem 1.1],
there exists a positive constant $c^{\prime}$ such that for all
$x,y\in{\mathbb{R}}$ with $|x-y|>1$,

$\displaystyle\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(y))\leq\mathrm{e}^{c^{\prime}|x-y|^{2}}\mathrm{e}^{-\frac{4}{3}|x-y|^{3}}.$
(5.9)

Thus, a combination of (5), (5) and (5.9) yields that for all
$\theta\in{\mathbb{R}}$,

$\displaystyle\left|\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]-\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}U_{k,N}}\right]\right|$
$\displaystyle\qquad\qquad\qquad\leq\frac{\theta^{2}}{2}[\text{\rm
Lip}(g_{N})]^{2}\sum_{\begin{subarray}{c}1\leq j,k\leq N\\\ j\neq
k\end{subarray}}\mathrm{e}^{c^{\prime}|x_{j,N}-x_{k,N}|^{2}}\,\mathrm{e}^{-\frac{4}{3}|x_{j,N}-x_{k,N}|^{3}}$
(5.10)

for all sufficiently large $N$.

Under the assumption (1.2), there exists $\alpha_{1}>0$ such that for all
sufficiently large $N$

$\displaystyle\min_{1\leq j\neq k\leq N}|x_{j,N}-x_{k,N}|\geq(3\log
N)^{1/3}(1+\alpha_{1}).$ (5.11)

Choose and fix $\epsilon\in(0,\frac{2}{3})$ such that

$\displaystyle 4(1+\alpha_{1})^{3}>2+\frac{2(2+6\epsilon)}{2+3\epsilon}.$

Furthermore, we can choose a sufficiently small number $\alpha_{2}\in(0,1)$
such that

$\displaystyle
3(\frac{4}{3}-\alpha_{2})(1+\alpha_{1})^{3}>2+\frac{2(2+6\epsilon)}{2+3\epsilon}.$
(5.12)

We first derive from (5) that for all sufficiently large $N$,

$\displaystyle\left|\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]-\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}U_{k,N}}\right]\right|$
$\displaystyle\leq\frac{\theta^{2}}{2}[\text{\rm
Lip}(g_{N})]^{2}\sum_{\begin{subarray}{c}1\leq j,k\leq N\\\ j\neq
k\end{subarray}}\mathrm{e}^{-(\frac{4}{3}-\alpha_{2})|x_{j,N}-x_{k,N}|^{3}}$
$\displaystyle\leq\frac{\theta^{2}}{2}[\text{\rm
Lip}(g_{N})]^{2}N^{2}\mathrm{e}^{-(\frac{4}{3}-\alpha_{2})\min_{1\leq j\neq
k\leq N}|x_{j,N}-x_{k,N}|^{3}}.$

Then, by (5.3) and Lemma 2.2, for all sufficiently large $N$,

$\displaystyle\left|\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]-\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}U_{k,N}}\right]\right|$
$\displaystyle\qquad\qquad\qquad\leq
C^{2}\theta^{2}N^{2}(N+1)^{\frac{2(2+6\epsilon)}{2+3\epsilon}}\mathrm{e}^{-(\frac{4}{3}-\alpha_{2})\min_{1\leq
j\neq k\leq N}|x_{j,N}-x_{k,N}|^{3}}$ $\displaystyle\qquad\qquad\qquad\leq
C^{2}\theta^{2}N^{2}(N+1)^{\frac{2(2+6\epsilon)}{2+3\epsilon}}N^{-3(\frac{4}{3}-\alpha_{2})(1+\alpha_{1})^{3}},\qquad\text{by
\eqref{min}}$

which imiplies by (5.12) that for all $\theta\in{\mathbb{R}}$,

$\displaystyle\lim_{N\to\infty}\left|\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}g_{N}(\mathcal{A}_{1}(x_{k,N}))}\right]-\mathrm{E}\left[\mathrm{e}^{{\rm
i}\theta\sum_{k=1}^{N}U_{k,N}}\right]\right|=0.$ (5.13)

Therefore, we complete the proof of (5.4) by (5.13) and (5.5). ∎

######  Proposition 5.2.

Fix $\lambda>0$. Let $g_{N-1}$ be defined as in (5.1) with $N$ replaced by
$N-1$ for $N\geq 2$. Let $\\{x_{k,N\\}_{N\geq 1,1\leq k\leq
N}}\subset{\mathbb{R}}$ satisfy the condition in (1.2). Then, as $N\to\infty$,

$\displaystyle\sum_{k=1}^{N}g_{N-1}(\mathcal{A}_{1}(x_{k,N}))\xrightarrow{\rm
d}{\rm Poisson}(\lambda).$ (5.14)

###### Proof.

The proof follows along the same lines as that of Proposition 5.1. One just
need replace $g_{N}$ by $g_{N-1}$. ∎

We are now ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

This is an immediate consequence of Propositions 5.1, 5.2 and (5.2). ∎

##  6 Asymptotic of maximum

In this section, we prove Theorems 1.4 and 1.5. We start with the following
estimate on the upper tail probability of the maximum of the Airy1 process.

######  Proposition 6.1.

There exists a positive constant $C$ such that for all $M\geq 2$

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>M\right\\}\leq
C\,M^{7/4}\mathrm{e}^{(2-\sqrt{2})M+\sqrt{2M}}\mathrm{e}^{-\frac{4}{3}\sqrt{2}M^{3/2}}.$
(6.1)

As a consequence, we have

$\displaystyle\lim_{M\to\infty}\frac{\log\mathrm{P}\left\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>M\right\\}}{M^{3/2}}=-\frac{4}{3}\sqrt{2}.$ (6.2)

###### Proof.

Because $\mathrm{P}\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>M\\}\geq\mathrm{P}\\{\mathcal{A}_{1}(0)>M\\}$, the
equality (6.2) follows immediately from the estimate (6.1) and the asymptotic
behavior of the GOE Tracy-Widom distribution (see [14, Theorem 1]). In order
to prove (6.1), we recall from [25, Theorem 4] that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>M\right\\}$
$\displaystyle=1-\mathrm{P}\left\\{\max_{0\leq x\leq 1}\mathcal{A}_{1}(x)\leq
M\right\\}$ $\displaystyle=1-\mathrm{P}\left\\{\mathcal{A}_{1}(x)\leq
M,\text{for all $x\in[0,1]$}\right\\}$
$\displaystyle=1-\det\left(I-B_{0}+\Lambda_{[0,1]}^{g}\mathrm{e}^{-\Delta}B_{0}\right)_{L^{2}({\mathbb{R}})},$
(6.3)

where in the Fredholm determinant, the kernel $\Lambda_{[0,1]}^{g}$ has the
formula given by (1.15) of [25] with $g\equiv M$ on $[0,1]$ and the kernel
$B_{0}$ is given in terms of the Airy function (see the equality below (1.5)
of [25]). We will adopt the method in the proof of [25, Proposition 2.3(a)] to
estimate the above tail probability.

We introduce the operator $U$ (depending on $M$) as follows

$\displaystyle Uf(x)=\mathrm{e}^{-\sqrt{2M}x}f(x).$

Let $\varphi(z)=\sqrt{1+z^{2}}$ and write

$\displaystyle V(x,z)$
$\displaystyle=\left(\mathrm{e}^{\Delta}-\Lambda_{[0,1]}^{g}\right)(x,z)\mathrm{e}^{-\sqrt{2M}x}\varphi(z)\mathrm{e}^{-\sqrt{2M}z},$
$\displaystyle W(z,y)$
$\displaystyle=\left(\mathrm{e}^{-\Delta}B_{0}\right)(z,y)\varphi(z)^{-1}\mathrm{e}^{\sqrt{2M}z}\mathrm{e}^{\sqrt{2M}y}.$

Then

$\displaystyle
U\left(B_{0}-\Lambda_{[0,1]}^{g}\mathrm{e}^{-\Delta}B_{0}\right)U^{-1}=VW.$

We will estimate the Hilbert-Schmidt norms $\|V\|_{2}$ and $\|W\|_{2}$ and the
use the inequality $\|VW\|_{1}\leq\|V\|_{2}\|W\|_{2}$ to give an upper bound
on the trace norm $\|VW\|_{1}$.

We first estimate $\|W\|_{2}$. Denote by ${\rm Ai}$ the Airy function and
$\|\varphi^{-1}\|_{2}$ the $L^{2}({\mathbb{R}})$-norm of $\varphi^{-1}$. Using
(1.9) of [25], we write

$\displaystyle\|W\|_{2}^{2}$
$\displaystyle=\int_{{\mathbb{R}}^{2}}\mathrm{d}x\mathrm{d}y\frac{\mathrm{e}^{-\frac{4}{3}+2(\sqrt{2M}-1)(x+y)}}{\varphi(x)^{2}}{\rm
Ai}(x+y+1)^{2}$
$\displaystyle=\|\varphi^{-1}\|_{2}^{2}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{4}{3}+2(\sqrt{2M}-1)y}{\rm
Ai}(y+1)^{2}\mathrm{d}y$
$\displaystyle=\|\varphi^{-1}\|_{2}^{2}\,\mathrm{e}^{-\frac{4}{3}-2(\sqrt{2M}-1)}\int_{-\infty}^{\infty}\mathrm{e}^{2(\sqrt{2M}-1)y}{\rm
Ai}(y)^{2}\mathrm{d}y.$

By the estimate (2.19) of [25] on the Airy function, we have for all $M\geq 2$

$\displaystyle\|W\|_{2}^{2}$ $\displaystyle\leq
C\|\varphi^{-1}\|_{2}^{2}\,\mathrm{e}^{-\frac{4}{3}-2(\sqrt{2M}-1)}\left(\int_{-\infty}^{0}\mathrm{e}^{2(\sqrt{2M}-1)y}\mathrm{d}y+\int_{0}^{\infty}\mathrm{e}^{2(\sqrt{2M}-1)y-\frac{4}{3}y^{3/2}}\mathrm{d}y\right)$
$\displaystyle\leq
c_{1}^{\prime}(\sqrt{2M}-1)^{2}\mathrm{e}^{-2(\sqrt{2M}-1)}\mathrm{e}^{\frac{2}{3}(\sqrt{2M}-1)^{3}},$
(6.4)

where the second inequality holds by Lemma A.1.

We proceed to estimate $\|V\|_{2}$. We use the formula (1.15) of [25] to write

$\displaystyle
V(x,y)=\frac{\varphi(y)}{\sqrt{4\pi}}\mathrm{e}^{-\frac{(x-y)^{2}}{4}-\sqrt{2M}(x+y)}\mathrm{P}_{\hat{b}(0)=x,\hat{b}(1)=y}\left(\hat{b}(t)\geq
M\,\,\text{for some $t\in[0,1]$}\right),$

where $\hat{b}$ denotes a Brownian bridge from $x$ at time $0$ to $y$ at time
$1$ with diffusion constant $2$. This probability equals
$\mathrm{e}^{-(x-M)(y-M)}$ if $x\leq M,y\leq M$ and $1$ otherwise (see line
one on page 621 of [25] or page 69 in [5]). Hence,

$\displaystyle\|V\|_{2}^{2}$
$\displaystyle=\frac{1}{4\pi}\int_{(-\infty,M]^{2}}\mathrm{d}x\mathrm{d}y\,(1+y^{2})\left[\mathrm{e}^{-\frac{(x-y)^{2}}{4}-\sqrt{2M}(x+y)-(x-M)(y-M)}\right]^{2}$
$\displaystyle\quad+\frac{1}{4\pi}\int_{{\mathbb{R}}^{2}\setminus(-\infty,M]^{2}}\mathrm{d}x\mathrm{d}y\,(1+y^{2})\left[\mathrm{e}^{-\frac{(x-y)^{2}}{4}-\sqrt{2M}(x+y)}\right]^{2}$
$\displaystyle:=I_{1}+I_{2}+I_{3}+I_{4},$

where

$\displaystyle I_{1}$
$\displaystyle=\frac{1}{4\pi}\int_{(-\infty,M]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\sqrt{2M}(x+y)-2(x-M)(y-M)},$
$\displaystyle I_{2}$
$\displaystyle=\frac{1}{4\pi}\int_{(-\infty,M]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\sqrt{2M}(x+y)-2(x-M)(y-M)},$
$\displaystyle I_{3}$
$\displaystyle=\frac{1}{4\pi}\int_{{\mathbb{R}}^{2}\setminus(-\infty,M]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\sqrt{2M}(x+y)},$
$\displaystyle I_{4}$
$\displaystyle=\frac{1}{4\pi}\int_{{\mathbb{R}}^{2}\setminus(-\infty,M]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\sqrt{2M}(x+y)}.$

Using change of variable,

$\displaystyle I_{1}$
$\displaystyle=\frac{\mathrm{e}^{-4\sqrt{2}M^{3/2}}}{4\pi}\int_{(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x+y)^{2}}{2}-2\sqrt{2M}(x+y)}$
$\displaystyle\leq
c_{2}^{\prime}\mathrm{e}^{8M}\mathrm{e}^{-4\sqrt{2}M^{3/2}}$ (6.5)

for all $M\geq 2$, where the inequality holds by Lemma A.2. Similarly, using
the inequality $(a+b)^{2}\leq 2a^{2}+2b^{2}$,

$\displaystyle I_{2}$
$\displaystyle=\frac{\mathrm{e}^{-4\sqrt{2}M^{3/2}}}{4\pi}\int_{(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,(y+M)^{2}\mathrm{e}^{-\frac{(x+y)^{2}}{2}-2\sqrt{2M}(x+y)}$
$\displaystyle\leq
2M^{2}I_{1}+\frac{\mathrm{e}^{-4\sqrt{2}M^{3/2}}}{2\pi}\int_{(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x+y)^{2}}{2}-2\sqrt{2M}(x+y)}$
$\displaystyle\leq
c_{3}^{\prime}M^{2}\mathrm{e}^{8M}\mathrm{e}^{-4\sqrt{2}M^{3/2}}$ (6.6)

for all $M\geq 2$, where the second inequality holds by (6) and Lemma A.3.

We move on to estimate $I_{3}$ and $I_{4}$. We use change of variable to see
that

$\displaystyle I_{3}$
$\displaystyle=\frac{\mathrm{e}^{-4\sqrt{2}M^{3/2}}}{4\pi}\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\sqrt{2M}(x+y)}$
$\displaystyle\leq
c_{4}^{\prime}\sqrt{M}\mathrm{e}^{4M}\mathrm{e}^{-4\sqrt{2}M^{3/2}}$ (6.7)

for all $M\geq 2$, where the inequality holds by Lemma A.4. It remains to
estimate $I_{4}$. By change of variable,

$\displaystyle I_{4}$
$\displaystyle=\frac{\mathrm{e}^{-4\sqrt{2}M^{3/2}}}{4\pi}\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,(y+M)^{2}\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\sqrt{2M}(x+y)}$
$\displaystyle\leq
2M^{2}I_{3}+\frac{\mathrm{e}^{-4\sqrt{2}M^{3/2}}}{2\pi}\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\sqrt{2M}(x+y)}$
$\displaystyle\leq
c_{5}^{\prime}M^{5/2}\mathrm{e}^{4M}\mathrm{e}^{-4\sqrt{2}M^{3/2}}$ (6.8)

for all $M\geq 2$, where the second inequality holds by (6) and Lemma A.4.

Now we combine the estimates (6), (6), (6),(6) and (6) to see that there exist
a constant $C^{\prime}>0$ such that for all $M\geq 2$,

$\displaystyle\|V\|_{2}^{2}\|W\|_{2}^{2}$ $\displaystyle\leq
C^{\prime}\,M^{7/2}\mathrm{e}^{(8-2\sqrt{2})M}\mathrm{e}^{\frac{2}{3}(\sqrt{2M}-1)^{3}-4\sqrt{2}M^{3/2}}$
$\displaystyle=C^{\prime}\,\mathrm{e}^{-2/3}M^{7/2}\mathrm{e}^{(4-2\sqrt{2})M+2\sqrt{2M}}\mathrm{e}^{-\frac{8}{3}\sqrt{2}M^{3/2}},$

which implies that for all $M\geq 2$,

$\displaystyle\|VW\|_{1}\leq\|V\|_{2}\|W\|_{2}\leq\sqrt{C^{\prime}}\mathrm{e}^{-1/3}M^{7/4}\mathrm{e}^{(2-\sqrt{2})M+\sqrt{2M}}\mathrm{e}^{-\frac{4}{3}\sqrt{2}M^{3/2}}.$
(6.9)

Thus, $U(B_{0}-\Lambda_{[0,1]}^{g}\mathrm{e}^{-\Delta}B_{0})U^{-1}$ is a trace
class operator on $L^{2}({\mathbb{R}})$ and we deduce from (6) that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>M\right\\}$
$\displaystyle=1-\det\left(I-U(B_{0}-\Lambda_{[0,1]}^{g}\mathrm{e}^{-\Delta}B_{0})U^{-1}\right)_{L^{2}({\mathbb{R}})}$
$\displaystyle=1-\det\left(I-VW\right)_{L^{2}({\mathbb{R}})}$
$\displaystyle\leq\|VW\|_{1}\mathrm{e}^{\|VW\|_{1}+1},$ (6.10)

where the inequality follows from (3.1) of [25]. Therefore, we conclude from
(6.9) and (6) that there exists $C>0$ such that for all $M\geq 2$,

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>M\right\\}\leq
C\,M^{7/4}\mathrm{e}^{(2-\sqrt{2})M+\sqrt{2M}}\mathrm{e}^{-\frac{4}{3}\sqrt{2}M^{3/2}}.$

The proof is complete. ∎

Before we prove Theorem 1.4, we present a probability inequality for the Airy1
process: there exists a constant $K>0$ such that for all $x,y\in{\mathbb{R}}$,

$\displaystyle\sup_{s,t\in{\mathbb{R}}}\left(\mathrm{P}\left\\{\mathcal{A}_{1}(x)\leq
s,\mathcal{A}_{1}(y)\leq t\right\\}-\mathrm{P}\left\\{\mathcal{A}_{1}(x)\leq
s\right\\}\mathrm{P}\left\\{\mathcal{A}_{1}(y)\leq t\right\\}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq
K\left[\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x)\,,\mathcal{A}_{1}(y))\right]^{1/3}.$ (6.11)

The above inequality follows from [24, (6.2.20)] (see also [24, Theorem
6.2.15]) because the Airy1 process is associated and its one-point
distribution, i.e., the GOE Tracy-Widom distribution has bounded and
continuous density and finite second moment.

We are now ready to prove Theorem 1.4.

###### Proof of Theorem 1.4.

We first show that almost surely

$\displaystyle\liminf_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)}{(\log
N)^{2/3}}\geq\frac{1}{2}\left(\frac{3}{2}\right)^{2/3}.$ (6.12)

Let $\beta$ be a positive number that is strictly less than $\frac{3}{2}$.
Choose and fix $a,\epsilon\in(0,1)$ ($a$ close to $1$ and $\epsilon$ close to
$0$) such that

$\displaystyle\beta<\frac{3a}{2(1+\epsilon)}.$ (6.13)

Define $x_{j}=jN/\left\lfloor N^{a}\right\rfloor$ for $j=1,\ldots,\left\lfloor
N^{a}\right\rfloor$. Here the notation $\left\lfloor x\right\rfloor$ denotes
the largest integer which is less than or equal to $x$. By the stationarity of
the Airy1 process,

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}\leq\mathrm{P}\left\\{\max_{1\leq j\leq\left\lfloor
N^{a}\right\rfloor}\mathcal{A}_{1}(x_{j})\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}$ $\displaystyle\qquad\qquad=\mathrm{P}\left\\{\max_{1\leq
j\leq\left\lfloor
N^{a}\right\rfloor}\mathcal{A}_{1}(x_{j})\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}-\prod_{j=1}^{\left\lfloor
N^{a}\right\rfloor}\mathrm{P}\left\\{\mathcal{A}_{1}(x_{j})\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\left(\mathrm{P}\left\\{\mathcal{A}_{1}(0)\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}\right)^{\left\lfloor N^{a}\right\rfloor}.$

Since the Airy1 process is associated, by (2.1), we see that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)\leq\frac{1}{2}(\beta\log N)^{2/3}\right\\}$
$\displaystyle\qquad\leq\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\bigg{(}\mathrm{P}\left\\{\mathcal{A}_{1}(x_{j})\leq\frac{1}{2}(\beta\log
N)^{2/3},\mathcal{A}_{1}(x_{k})\leq\frac{1}{2}(\beta\log N)^{2/3}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\mathrm{P}\left\\{\mathcal{A}_{1}(x_{j})\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}\mathrm{P}\left\\{\mathcal{A}_{1}(x_{k})\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}\bigg{)}$
$\displaystyle\qquad\qquad+\left(\mathrm{P}\left\\{\mathcal{A}_{1}(0)\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}\right)^{\left\lfloor N^{a}\right\rfloor}.$

We apply (6) to deduce that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)\leq\frac{1}{2}(\beta\log N)^{2/3}\right\\}$
$\displaystyle\qquad\leq K\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\left[\operatorname{\text{\rm
Cov}}(\mathcal{A}_{1}(x_{j})\,,\mathcal{A}_{1}(x_{k}))\right]^{1/3}+\left(\mathrm{P}\left\\{\mathcal{A}_{1}(0)\leq\frac{1}{2}(\beta\log
N)^{2/3}\right\\}\right)^{\left\lfloor N^{a}\right\rfloor}.$

We can assume that $N$ is sufficiently large such that $N/\left\lfloor
N^{a}\right\rfloor>\max\\{1,3c^{\prime}\\}$ with $c^{\prime}$ being the number
in (5.9), $\left\lfloor N^{a}\right\rfloor>N^{a}/2$ and $(\beta\log
N)^{2/3}>C_{2}$, where $C_{2}$ is the constant in (2.12). Hence, by (5.9),

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)\leq\frac{1}{2}(\beta\log N)^{2/3}\right\\}$
$\displaystyle\qquad\qquad\qquad\leq K\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\mathrm{e}^{\frac{c^{\prime}}{3}|x_{j}-x_{k}|^{2}}\mathrm{e}^{-\frac{4}{9}|x_{j}-x_{k}|^{3}}+F_{1}\left((\beta\log
N)^{2/3}\right)^{\left\lfloor N^{a}\right\rfloor}$
$\displaystyle\qquad\qquad\qquad\leq K\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\mathrm{e}^{-\frac{1}{3}|x_{j}-x_{k}|^{3}}+\left(1-\mathrm{e}^{-\frac{2}{3}(1+\epsilon)\beta\log
N}\right)^{\left\lfloor N^{a}\right\rfloor}$
$\displaystyle\qquad\qquad\qquad\leq
KN^{2a}\mathrm{e}^{-\frac{1}{3}N^{3(1-a)}}+\mathrm{e}^{-N^{-\frac{2}{3}(1+\epsilon)\beta}\left\lfloor
N^{a}\right\rfloor}\leq
KN^{2a}\mathrm{e}^{-\frac{1}{3}N^{3(1-a)}}+\mathrm{e}^{-\frac{1}{2}N^{-\frac{2}{3}(1+\epsilon)\beta+a}},$

where in the second inequality, we use the first inequality in (2.12), and in
the third inequality, we use the fact $1-x\leq\mathrm{e}^{-x}$ for all $x\geq
0$. Since $-\frac{2}{3}(1+\epsilon)\beta+a>0$ by (6.13), we conclude that

$\displaystyle\sum_{N=1}^{\infty}\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)\leq\frac{1}{2}(\beta\log N)^{2/3}\right\\}<\infty.$

Therefore, Borel-Cantelli’s lemma implies that almost surely,

$\displaystyle\liminf_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)}{(\log N)^{2/3}}\geq\frac{1}{2}\beta^{2/3}.$

The above liminf is taken along integers. By the monotonicity of $\max_{0\leq
x\leq N}\mathcal{A}_{1}(x)$ in $N$, we see that the preceding also holds when
the liminf is taken along natural numbers. Finally, by letting
$\beta\uparrow\frac{3}{2}$, we obtain (6.12).

We turn to proving that almost surely

$\displaystyle\limsup_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)}{(\log
N)^{2/3}}\leq\frac{1}{2}\left(\frac{3}{2}\right)^{2/3}.$ (6.14)

Let $\gamma$ be a positive number that is strictly larger than
$\frac{3}{4\sqrt{2}}$. Choose and fix $\epsilon\in(0,1)$ such that

$\displaystyle\gamma>\frac{1}{\frac{4}{3}\sqrt{2}-\epsilon}.$ (6.15)

For this fixed $\epsilon\in(0,1)$, Proposition 6.1 ensures that there exists
constant $C_{1}>0$ depending on $\epsilon$ such that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>M\right\\}\leq\mathrm{e}^{-(\frac{4}{3}\sqrt{2}-\epsilon)M^{3/2}},\quad\text{for
all $M\geq C_{1}$}.$ (6.16)

Assume for the moment that $N$ is a sufficiently large integer such that
$(\gamma\log N)^{2/3}\geq C_{1}$ with $C_{1}$ being the constant in (6.16). By
the stationarity of the Airy1 process,

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)>(\gamma\log N)^{2/3}\right\\}$
$\displaystyle=\mathrm{P}\left\\{\cup_{j=1}^{N}\left\\{\max_{j-1\leq x\leq
j}\mathcal{A}_{1}(x)>(\gamma\log N)^{3/2}\right\\}\right\\}$
$\displaystyle\leq N\mathrm{P}\left\\{\max_{0\leq x\leq
1}\mathcal{A}_{1}(x)>(\gamma\log N)^{2/3}\right\\}$ $\displaystyle\leq
N\mathrm{e}^{-(\frac{4}{3}\sqrt{2}-\epsilon)\gamma\log
N}=N^{1-(\frac{4}{3}\sqrt{2}-\epsilon)\gamma},$

where the second inequality follows from (6.16). Since
$(\frac{4}{3}\sqrt{2}-\epsilon)\gamma>1$ by (6.15), letting $N=2^{n}$, we
obtain that

$\displaystyle\sum_{n=1}^{\infty}\mathrm{P}\left\\{\max_{0\leq x\leq
2^{n}}\mathcal{A}_{1}(x)>(\gamma\log 2^{n})^{2/3}\right\\}<\infty,$

which implies that almost surely

$\displaystyle\limsup_{n\to\infty}\frac{\max_{0\leq x\leq
2^{n}}\mathcal{A}_{1}(x)}{(\log 2^{n})^{2/3}}\leq\gamma^{2/3}$

by Borel-Cantelli’s lemma. Since the quantity $\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)$ is monotone in $N$, we obtain that almost surely

$\displaystyle\limsup_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{1}(x)}{(\log N)^{2/3}}\leq\gamma^{2/3}.$

Letting $\gamma\downarrow\frac{3}{4\sqrt{2}}$, we prove (6.14).

Theorem 1.4 follows from (6.12) and (6.14). ∎

We proceed to prove Theorem 1.5. Recall from [14, Theorem 1] that for
$\epsilon\in(0,1)$, there exists a positive constant $\tilde{C}_{2}$ depending
on $\epsilon$ such that

$\displaystyle\mathrm{e}^{-(\frac{4}{3}+\epsilon)s^{3/2}}\leq\mathrm{P}\\{\mathcal{A}_{2}(0)\geq
s\\}\leq\mathrm{e}^{-(\frac{4}{3}-\epsilon)s^{3/2}},\quad\text{for all
$s\geq\tilde{C}_{2}$}.$ (6.17)

Analogous to (6), using the association property of the Airy2 process and the
fact that the GUE Tracy-Widom distribution has bounded continuous probability
density function and finite second moment, we see from [24, (6.2.20)] (see
also [24, Theorem 6.2.15]) that there exists a constant $K>0$ such that for
all $x,y\in{\mathbb{R}}$,

$\displaystyle\sup_{s,t\in{\mathbb{R}}}\left(\mathrm{P}\left\\{\mathcal{A}_{2}(x)\leq
s,\mathcal{A}_{2}(y)\leq t\right\\}-\mathrm{P}\left\\{\mathcal{A}_{2}(x)\leq
s\right\\}\mathrm{P}\left\\{\mathcal{A}_{2}(y)\leq t\right\\}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq
K\left[\operatorname{\text{\rm
Cov}}(\mathcal{A}_{2}(x)\,,\mathcal{A}_{2}(y))\right]^{1/3}.$ (6.18)

Now we start the proof of Theorem 1.5.

###### Proof of Theorem 1.5.

We first prove the lower bound in (1.5). The strategy is similar to that of
the proof of (1.4), using the association property of the Airy2 process, the
asymptotic behavior of the covariance of the Airy2 process and the tail
distribution of the GUE Tracy-Widom distribution.

Let $\beta$ be a positive number that is strictly less than $\frac{1}{4}$.
Choose and fix $a\in(0,\frac{1}{3})$, $\epsilon\in(0,1)$ such that

$\displaystyle\beta<\frac{a}{\frac{4}{3}+\epsilon}.$ (6.19)

Define $x_{j}=jN/\left\lfloor N^{a}\right\rfloor$ for $j=1,\ldots,\left\lfloor
N^{a}\right\rfloor$. Then by the stationarity of the Airy2 process,

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{2}(x)\leq(\beta\log
N)^{2/3}\right\\}\leq\mathrm{P}\left\\{\max_{1\leq j\leq\left\lfloor
N^{a}\right\rfloor}\mathcal{A}_{2}(x_{j})\leq(\beta\log N)^{2/3}\right\\}$
$\displaystyle\qquad\qquad=\mathrm{P}\left\\{\max_{1\leq j\leq\left\lfloor
N^{a}\right\rfloor}\mathcal{A}_{2}(x_{j})\leq(\beta\log
N)^{2/3}\right\\}-\prod_{j=1}^{\left\lfloor
N^{a}\right\rfloor}\mathrm{P}\left\\{\mathcal{A}_{2}(x_{j})\leq(\beta\log
N)^{2/3}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\left(\mathrm{P}\left\\{\mathcal{A}_{2}(0)\leq(\beta\log
N)^{2/3}\right\\}\right)^{\left\lfloor N^{a}\right\rfloor}.$

We can assume that $N$ is sufficiently large such that $\left\lfloor
N^{a}\right\rfloor>N^{a}/2$ and $(\beta\log N)^{2/3}>C^{\prime}_{2}$, where
$C^{\prime}_{2}$ is the constant in (6.17). Since the Airy2 process is
associated (see Proposition 3.2), by (2.1), we see that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{2}(x)\leq(\beta\log N)^{2/3}\right\\}$
$\displaystyle\qquad\leq\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\bigg{(}\mathrm{P}\left\\{\mathcal{A}_{2}(x_{j})\leq(\beta\log
N)^{2/3},\mathcal{A}_{2}(x_{k})\leq(\beta\log N)^{2/3}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\mathrm{P}\left\\{\mathcal{A}_{2}(x_{j})\leq(\beta\log
N)^{2/3}\right\\}\mathrm{P}\left\\{\mathcal{A}_{2}(x_{k})\leq(\beta\log
N)^{2/3}\right\\}\bigg{)}$
$\displaystyle\qquad\qquad+\left(1-\mathrm{P}\left\\{\mathcal{A}_{2}(0)>(\beta\log
N)^{2/3}\right\\}\right)^{\left\lfloor N^{a}\right\rfloor}$
$\displaystyle\qquad\leq K\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\left[\operatorname{\text{\rm
Cov}}(\mathcal{A}_{2}(x_{j})\,,\mathcal{A}_{2}(x_{k}))\right]^{1/3}+\left(1-\mathrm{P}\left\\{\mathcal{A}_{2}(0)>(\beta\log
N)^{2/3}\right\\}\right)^{\left\lfloor N^{a}\right\rfloor},$

where the second inequality holds by (6). Moreover, since the decay rate of
the covariance $\operatorname{\text{\rm
Cov}}(\mathcal{A}_{2}(x)\,,\mathcal{A}_{2}(0))$ is $x^{-2}$ as $x\to\infty$
(see [32]), we deduce that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{2}(x)\leq(\beta\log N)^{2/3}\right\\}$ $\displaystyle\leq
K^{\prime}\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\frac{1}{|x_{k}-x_{j}|^{2/3}}+\left(1-\mathrm{e}^{-(\frac{4}{3}+\epsilon)\beta\log
N}\right)^{\left\lfloor N^{a}\right\rfloor}$ $\displaystyle\leq
K^{\prime}N^{\frac{2}{3}(a-1)}\sum_{1\leq j<k\leq\left\lfloor
N^{a}\right\rfloor}\frac{1}{|k-j|^{2/3}}+\mathrm{e}^{-\left\lfloor
N^{a}\right\rfloor N^{-(\frac{4}{3}+\epsilon)\beta}},$

where we have used (6.17) in the first inequality and $1-x\leq\mathrm{e}^{-x}$
for all $x\geq 0$ in the second inequality. Because for integer $m\geq 2$,

$\displaystyle\sum_{k=2}^{m}\sum_{j=1}^{k-1}\frac{1}{(k-j)^{2/3}}$
$\displaystyle=\sum_{k=2}^{m}\sum_{j=1}^{k-1}\frac{1}{j^{2/3}}\leq\sum_{k=2}^{m}\int_{0}^{k-1}\frac{\mathrm{d}x}{x^{2/3}}=3\sum_{k=2}^{m}(k-1)^{1/3}$
$\displaystyle\leq\frac{9}{4}\int_{0}^{m}x^{1/3}\mathrm{d}x=\frac{9}{4}m^{4/3},$

we conclude that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
N}\mathcal{A}_{2}(x)\leq(\beta\log N)^{2/3}\right\\}$
$\displaystyle\leq\frac{9K^{\prime}}{4}N^{\frac{2}{3}(a-1)+\frac{4}{3}a}+\mathrm{e}^{-\frac{1}{2}N^{a-(\frac{4}{3}+\epsilon)\beta}}.$

Letting $N=2^{n}$, we obtain that

$\displaystyle\mathrm{P}\left\\{\max_{0\leq x\leq
2^{n}}\mathcal{A}_{2}(x)\leq(\beta\log 2^{n})^{2/3}\right\\}$
$\displaystyle\leq\frac{9K^{\prime}}{4}2^{-\frac{2}{3}(1-3a)n}+\mathrm{e}^{-\frac{1}{2}2^{(a-(\frac{4}{3}+\epsilon)\beta)n}}.$

Since $a<\frac{1}{3}$ and $a>(\frac{4}{3}+\epsilon)\beta$ (see (6.19)), we see
that

$\displaystyle\sum_{n=1}^{\infty}\mathrm{P}\left\\{\max_{0\leq x\leq
2^{n}}\mathcal{A}_{2}(x)\leq(\beta\log 2^{n})^{2/3}\right\\}<\infty.$

Hence, by Borel-Cantelli’s lemma, we have almost surely,

$\displaystyle\liminf_{n\to\infty}\frac{\max_{0\leq x\leq
2^{n}}\mathcal{A}_{2}(x)}{(\log 2^{n})^{2/3}}\geq\beta^{2/3}.$

A monotonicity argument yields that almost surely,

$\displaystyle\liminf_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{2}(x)}{(\log N)^{2/3}}\geq\beta^{2/3}.$

Letting $\beta\uparrow\frac{1}{4}$, we obtain the lower bound in (1.5).

We proceed to prove the upper bound in (1.5). By [8, Corollary 1.3] and
stationarity of the Airy2 process, there exists a positive constant $s_{0}$
such that

$\displaystyle\sup_{x\in{\mathbb{R}}}\mathrm{P}\left(\sup_{0\leq y\leq
2}|\mathcal{A}_{2}(x+y)-\mathcal{A}_{2}(x)|\geq
s\right)\leq\mathrm{e}^{-\frac{s^{2}}{16}},\quad\text{for all $s>s_{0}$}.$
(6.20)

Let $\gamma$ be a positive number that is strictly larger than $\frac{3}{4}$.
Choose and fix $\delta\in(0,1)$ and $\epsilon\in(0,1)$ such that

$\displaystyle(\frac{4}{3}-\epsilon)\delta^{\frac{3}{2}}\gamma>1.$ (6.21)

Assume $N$ is sufficiently large so that

$\displaystyle\delta(\gamma\log
N)^{2/3}>\tilde{C}_{2}\quad\text{and}\quad(1-\delta)(\gamma\log
N)^{2/3}>s_{0},$

where the constants $\tilde{C}_{2}$ and $s_{0}$ are given in (6.17) and (6.20)
respectively. By the triangle inequality and stationarity of the Airy2
process,

$\displaystyle\mathrm{P}\left(\max_{0\leq x\leq
2N}\mathcal{A}_{2}(x)\geq(\gamma\log(2N))^{2/3}\right)\leq\mathrm{P}\left(\cup_{j=0}^{N-1}\left\\{\mathcal{A}_{2}(2j)\geq\delta(\gamma\log(2N))^{2/3}\right\\}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\mathrm{P}\left(\cup_{j=1}^{N}\sup_{y\in[2(j-1),2j]}|\mathcal{A}_{2}(y)-\mathcal{A}_{2}(2(j-1))|\geq(1-\delta)(\gamma\log(2N))^{2/3}\right)$
$\displaystyle\qquad\quad\leq
N\mathrm{P}\left(\mathcal{A}_{2}(0)\geq\delta(\gamma\log(2N))^{2/3}\right)+N\mathrm{P}\left(\sup_{y\in[0,\,2]}|\mathcal{A}_{2}(y)-\mathcal{A}_{2}(0)|\geq(1-\delta)(\gamma\log(2N))^{2/3}\right)$
$\displaystyle\qquad\quad\leq
N\mathrm{e}^{-(\frac{4}{3}-\epsilon)\delta^{\frac{3}{2}}\gamma\log(2N)}+N\mathrm{e}^{-\frac{(1-\delta)^{2}}{16}(\gamma\log(2N))^{4/3}}$
$\displaystyle\qquad\quad=c_{0}N^{-(\frac{4}{3}-\epsilon)\delta^{\frac{3}{2}}\gamma+1}+N\mathrm{e}^{-\frac{(1-\delta)^{2}}{16}(\gamma\log(2N))^{4/3}},$

where the last inequality follows from (6.17) and (6.20). Letting $N=2^{n}$
and because of (6.21), we obtain that

$\displaystyle\sum_{n=1}^{\infty}\mathrm{P}\left(\max_{0\leq x\leq
2^{n+1}}\mathcal{A}_{2}(x)\geq(\gamma\log 2^{n+1})^{2/3}\right)<\infty,$

which implies that almost surely,

$\displaystyle\limsup_{n\to\infty}\frac{\max_{0\leq x\leq
2^{n+1}}\mathcal{A}_{2}(x)}{\log 2^{n+1}}\leq\gamma^{2/3},$

thanks to Borel-Cantelli’s lemma. Appealing to the monotonicity argument
again, we conclude that almost surely,

$\displaystyle\limsup_{N\to\infty}\frac{\max_{0\leq x\leq
N}\mathcal{A}_{2}(x)}{\log N}\leq\gamma^{2/3}.$

Letting $\gamma\downarrow\frac{3}{4}$, we obtain the upper bound in (1.5).

The proof is complete. ∎

##  Appendix A Appendix

We include in this section a few technical lemmas which are used in the proof
of Proposition 6.1.

######  Lemma A.1.

There exists a constant $c_{1}>0$ such that for all $\theta\geq 1$

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{2\theta
y-\frac{4}{3}y^{3/2}}\mathrm{d}y\leq
c_{1}\theta^{2}\mathrm{e}^{\frac{2}{3}\theta^{3}}.$ (A.1)

###### Proof.

The estimate follows from a standard calculation in the steepest-descent
method (see for instance [2, Appendix B]). We first use change of variable to
see that

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{2\theta
y-\frac{4}{3}y^{3/2}}\mathrm{d}y=\theta^{2}\int_{0}^{\infty}\mathrm{e}^{\theta^{3}(2y-\frac{4}{3}y^{3/2})}\mathrm{d}y.$

Denote $h(y)=2y-\frac{4}{3}y^{3/2}$. It is clear that the function $h$ is
increasing on $[0,\frac{1}{2}]$, maximized at $1$ in $[\frac{1}{2},9]$ and
$h(y)\leq-2y$ for all $y\geq 9$. Hence,

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{\theta^{3}(2y-\frac{3}{4}y^{3/2})}\mathrm{d}y$
$\displaystyle=\int_{0}^{\frac{1}{2}}\mathrm{e}^{\theta^{3}(2y-\frac{3}{4}y^{3/2})}\mathrm{d}y+\int_{\frac{1}{2}}^{9}\mathrm{e}^{\theta^{3}(2y-\frac{3}{4}y^{3/2})}\mathrm{d}y+\int_{9}^{\infty}\mathrm{e}^{-2\theta^{3}y}\mathrm{d}y$
$\displaystyle\leq\frac{1}{2}\mathrm{e}^{h(1/2)\theta^{3}}+\frac{17}{2}\mathrm{e}^{h(1)\theta^{3}}+\frac{1}{2\theta^{3}}\mathrm{e}^{-18\theta^{3}}$
$\displaystyle\leq
c_{1}\mathrm{e}^{h(1)\theta^{3}}=c_{1}\mathrm{e}^{\frac{2}{3}\theta^{3}}.$

This proves (A.1). ∎

######  Lemma A.2.

There exists a constant $c_{2}>0$ such that for all $\theta\geq 1$,

$\displaystyle\int_{(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x+y)^{2}}{2}-2\theta(x+y)}\leq
c_{2}\,\mathrm{e}^{4\theta^{2}}.$ (A.2)

###### Proof.

Using change of variable,

$\displaystyle\int_{(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x+y)^{2}}{2}-2\theta(x+y)}=\theta^{2}\int_{[0,\infty)^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}$
$\displaystyle\qquad\qquad\qquad\qquad\leq\theta^{2}\int_{[0,4]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}+2\theta^{2}\int_{0}^{4}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\theta^{2}\int_{[4,\infty)^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}.$

It is clear that

$\displaystyle\int_{[0,4]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}\leq
16\mathrm{e}^{2\theta^{2}}.$

Moreover,

$\displaystyle\int_{0}^{4}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}\leq
4\int_{-\infty}^{\infty}\mathrm{e}^{\theta^{2}(-\frac{y^{2}}{2}+2y)}\mathrm{d}y=\frac{4}{\theta}\mathrm{e}^{2\theta^{2}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{1}{2}y^{2}}\mathrm{d}y.$

Furthermore,

$\displaystyle\int_{[4,\infty)^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}\leq\left(\int_{4}^{\infty}\mathrm{e}^{\theta^{2}(-\frac{y^{2}}{2}+2y)}\mathrm{d}y\right)^{2}\leq\theta^{-2}\mathrm{e}^{4\theta^{2}}\left(\int_{-\infty}^{\infty}\mathrm{e}^{-y^{2}/2}\mathrm{d}y\right)^{2}.$

The preceding estimates prove (A.2). ∎

######  Lemma A.3.

There exists a constant $c_{3}>0$ such that for all $\theta\geq 1$,

$\displaystyle\int_{(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x+y)^{2}}{2}-2\theta(x+y)}\leq
c_{3}\,\theta^{2}\mathrm{e}^{4\theta^{2}}.$ (A.3)

###### Proof.

The proof is similar to that of Lemma A.2. Using change of variable,

$\displaystyle\int_{(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x+y)^{2}}{2}-2\theta(x+y)}=\theta^{4}\int_{[0,\infty)^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}$
$\displaystyle\qquad\qquad\leq\theta^{4}\int_{[0,4]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}+2\theta^{4}\int_{0}^{4}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,(x+y)^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}$
$\displaystyle\qquad\qquad\quad+\theta^{4}\int_{[4,\infty)^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}.$

Evidently,

$\displaystyle\int_{[0,4]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}\leq
256\mathrm{e}^{2\theta^{2}}.$

Moreover,

$\displaystyle\int_{0}^{4}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,(x+y)^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}\leq
4\int_{-\infty}^{\infty}y^{2}\mathrm{e}^{\theta^{2}(-\frac{y^{2}}{2}+2y)}\mathrm{d}y$
$\displaystyle\qquad\qquad\qquad\qquad\leq\frac{8}{\theta^{3}}\mathrm{e}^{2\theta^{2}}\int_{-\infty}^{\infty}y^{2}\mathrm{e}^{-\frac{y^{2}}{2}}\mathrm{d}y+\frac{32}{\theta}\mathrm{e}^{2\theta^{2}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{y^{2}}{2}}\mathrm{d}y.$

Furthermore,

$\displaystyle\int_{[4,\infty)^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x+y))}$
$\displaystyle\leq\int_{-\infty}^{\infty}\mathrm{e}^{\theta^{2}(-\frac{x^{2}}{2}+2x)}\mathrm{d}x\int_{-\infty}^{\infty}y^{2}\mathrm{e}^{\theta^{2}(-\frac{y^{2}}{2}+2y)}\mathrm{d}y$
$\displaystyle\leq\frac{C}{\theta^{2}}\mathrm{e}^{4\theta^{2}}$

for some constant $C>0$. The preceding estimates prove (A.3). ∎

######  Lemma A.4.

There exists a constant $c_{4}>0$ such that for all $\theta\geq 1$,

$\displaystyle\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\theta(x+y)}\leq
c_{4}\,\theta\mathrm{e}^{2\theta^{2}}.$ (A.4)

###### Proof.

Using change of variable,

$\displaystyle\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\theta(x+y)}$
$\displaystyle=\theta^{2}\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x-y)^{2}}{2}-2(x+y))}$
$\displaystyle:=\theta^{2}(J_{1}+2J_{2}),$

where

$\displaystyle J_{1}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x-y)^{2}}{2}-2(x+y))},$
$\displaystyle J_{2}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\int_{-\infty}^{0}\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x-y)^{2}}{2}-2(x+y))}.$

We perform change of variables $x+y=u,x-y=v$ to see that

$\displaystyle J_{1}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}u\int_{-u}^{u}\mathrm{d}v\,\mathrm{e}^{\theta^{2}(-\frac{v^{2}}{2}-2u)}\leq\int_{0}^{\infty}u\mathrm{e}^{-2\theta^{2}u}\mathrm{d}u=\frac{1}{\theta^{4}}\int_{0}^{\infty}u\mathrm{e}^{-2u}\mathrm{d}u.$

Similarly,

$\displaystyle J_{2}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}-2(x-y))}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}u\int_{-u}^{u}\mathrm{d}v\,\mathrm{e}^{\theta^{2}(-\frac{u^{2}}{2}-2v)}\leq\int_{0}^{\infty}u\mathrm{e}^{\theta^{2}(-\frac{u^{2}}{2}+2u)}\mathrm{d}u$
$\displaystyle\leq\mathrm{e}^{2\theta^{2}}\int_{-\infty}^{\infty}|u|\mathrm{e}^{\theta^{2}(u-2)^{2}/2}\mathrm{d}u\leq\mathrm{e}^{2\theta^{2}}\left(\int_{-\infty}^{\infty}|u|\mathrm{e}^{\theta^{2}u^{2}/2}\mathrm{d}u+2\int_{-\infty}^{\infty}\mathrm{e}^{\theta^{2}u^{2}/2}\mathrm{d}u\right)$
$\displaystyle=\mathrm{e}^{2\theta^{2}}\left(\theta^{-2}\int_{-\infty}^{\infty}|u|\mathrm{e}^{u^{2}/2}\mathrm{d}u+2\theta^{-1}\int_{-\infty}^{\infty}\mathrm{e}^{u^{2}/2}\mathrm{d}u\right).$

The bounds on $J_{1},J_{2}$ prove (A.4). ∎

######  Lemma A.5.

There exists a constant $c_{5}>0$ such that for all $\theta\geq 1$,

$\displaystyle\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\theta(x+y)}\leq
c_{5}\,\theta^{3}\mathrm{e}^{2\theta^{2}}.$ (A.5)

###### Proof.

The proof is similar to that of Lemma A.4. We write

$\displaystyle\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{-\frac{(x-y)^{2}}{2}-2\theta(x+y)}$
$\displaystyle=\theta^{4}\int_{{\mathbb{R}}^{2}\setminus(-\infty,0]^{2}}\mathrm{d}x\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x-y)^{2}}{2}-2(x+y))}$
$\displaystyle:=\theta^{4}(T_{1}+T_{2}+T_{3}),$

where

$\displaystyle T_{1}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x-y)^{2}}{2}-2(x+y))},$
$\displaystyle T_{2}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\int^{0}_{-\infty}\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x-y)^{2}}{2}-2(x+y))},$
$\displaystyle T_{3}$
$\displaystyle=\int^{0}_{-\infty}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x-y)^{2}}{2}-2(x+y))}.$

Using change of variables $x+y=u,x-y=v$,

$\displaystyle T_{1}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}u\int_{-u}^{u}\mathrm{d}v\,\frac{(u-v)^{2}}{4}\mathrm{e}^{\theta^{2}(-\frac{v^{2}}{2}-2u)}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{\infty}u^{2}\mathrm{e}^{-2\theta^{2}u}\mathrm{d}u\int_{-\infty}^{\infty}\mathrm{e}^{-\theta^{2}v^{2}/2}\mathrm{d}v=\frac{1}{2\theta^{5}}\int_{0}^{\infty}u^{2}\mathrm{e}^{-2u}\mathrm{d}u\int_{-\infty}^{\infty}\mathrm{e}^{-v^{2}/2}\mathrm{d}v.$

Similarly,

$\displaystyle T_{2}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}-2(x-y))}=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}u\int_{-u}^{u}\mathrm{d}v\,\frac{(u-v)^{2}}{4}\mathrm{e}^{\theta^{2}(-\frac{u^{2}}{2}-2v)}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{\infty}\mathrm{d}u\int_{-u}^{u}\mathrm{d}v\,u^{2}\mathrm{e}^{\theta^{2}(-\frac{u^{2}}{2}-2v)}\leq\int_{0}^{\infty}u^{3}\mathrm{e}^{\theta^{2}(-\frac{u^{2}}{2}+2u)}\mathrm{d}u$
$\displaystyle\leq\mathrm{e}^{2\theta^{2}}\int_{-\infty}^{\infty}|u+2|^{3}\mathrm{e}^{-\theta^{2}u^{2}/2}\mathrm{d}u\leq
4\mathrm{e}^{2\theta^{2}}\int_{-\infty}^{\infty}(|u|^{3}+8)\mathrm{e}^{-\theta^{2}u^{2}/2}\mathrm{d}u$
$\displaystyle=4\mathrm{e}^{2\theta^{2}}\left(\theta^{-4}\int_{-\infty}^{\infty}|u|^{3}\mathrm{e}^{-u^{2}/2}\mathrm{d}u+8\theta^{-1}\int_{-\infty}^{\infty}\mathrm{e}^{-u^{2}/2}\mathrm{d}u\right),$

where in the fourth inequality, we used the inequality $(|a|+|b|)^{3}\leq
4(|a|^{3}+|b^{3}|)$. Moreover,

$\displaystyle T_{3}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\int_{0}^{\infty}\mathrm{d}y\,y^{2}\mathrm{e}^{\theta^{2}(-\frac{(x+y)^{2}}{2}+2(x-y))}=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}u\int_{-u}^{u}\mathrm{d}v\,\frac{(u-v)^{2}}{4}e^{\theta^{2}(-\frac{u^{2}}{2}+2v)}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{\infty}\mathrm{d}u\int_{-u}^{u}\mathrm{d}v\,u^{2}e^{\theta^{2}(-\frac{u^{2}}{2}+2v)}\leq\int_{0}^{\infty}u^{3}\mathrm{e}^{\theta^{2}(-\frac{u^{2}}{2}+2u)}\mathrm{d}u$
$\displaystyle\leq
4\mathrm{e}^{2\theta^{2}}\left(\theta^{-4}\int_{-\infty}^{\infty}|u|^{3}\mathrm{e}^{-u^{2}/2}\mathrm{d}u+8\theta^{-1}\int_{-\infty}^{\infty}\mathrm{e}^{-u^{2}/2}\mathrm{d}u\right),$

where the last inequality holds by the same arguments as in the estimate of
$T_{2}$. The bounds on $T_{1},T_{2},T_{3}$ prove (A.5). ∎

Acknowledgement. Research supported in part by Beijing Natural Science
Foundation (No. 1232010), National Natural Science Foundation of China (No.
12201047) and National Key R&D Program of China (No. 2022YFA 1006500). The
author would like to thank Davar Khoshnevisan and Jeremy Quastel for useful
discussions.

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Fei Pu Laboratory of Mathematics and Complex Systems, School of Mathematical
Sciences, Beijing Normal University, 100875, Beijing, China.
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