# Universality for multiplicative statistics of Hermitian random matrices and
the integro-differential Painlevé II equation

Promit Ghosal  Department of Mathematics, Massachusetts Institute of
Technology, USA<EMAIL_ADDRESS>and  Guilherme L. F. Silva  Instituto de
Ciências Matemáticas e de Computação, Universidade de São Paulo (ICMC - USP),
São Carlos, São Paulo, Brazil<EMAIL_ADDRESS>
###### Abstract.

We study multiplicative statistics for the eigenvalues of unitarily-invariant
Hermitian random matrix models. We consider one-cut regular polynomial
potentials and a large class of multiplicative statistics. We show that in the
large matrix limit several associated quantities converge to limits which are
universal in both the potential and the family of multiplicative statistics
considered. In turn, such universal limits are described by the integro-
differential Painlevé II equation, and in particular they connect the random
matrix models considered with the narrow wedge solution to the KPZ equation at
any finite time.

###### Contents

  1. 1 Introduction
  2. 2 Statement of main results
  3. 3 About our approach: issues and extensions
  4. 4 A Model Riemann-Hilbert Problem
  5. 5 The RHP for the integro-differential RHP
  6. 6 Bounds on the id-PII RHP
  7. 7 Asymptotic analysis for the model problem with admissible data
  8. 8 The underlying equilibrium measure and related quantities
  9. 9 Associated orthogonal polynomials
  10. 10 The RHP analysis for the orthogonal polynomials
  11. 11 Proof of main results
  12. 12 Technical Lemmas
  13. A Laplace-type integrals

##  1\. Introduction

Random matrix theory has proven over time to be a powerful modern tool in
mathematics and physics. With widespread applications in different areas such
as engineering, statistical mechanics, probability, number theory, to mention
only a few, its theory is rich and has been under intense development in the
past thirty or so years. In a sense, much of the success of random matrix
theory has been due to its exact solvability, or integrability, turning them
into touchstones for predicting and confirming complex phenomenon in nature.

One of the most celebrated results in random matrix theory is the convergence
of fluctuations of the largest eigenvalue towards the Tracy-Widom law
$F_{\mathrm{GUE}}$. This result was first obtained by Tracy and Widom [58] for
matrices from the Gaussian Unitary Ensemble (GUE), who also showed that
$F_{\mathrm{GUE}}$ is expressible in terms of a particular solution to the
Painlevé II equation (shortly PII). Their findings sparked numerous advances
in mathematics and physics, which began from the extension to several other
matrix models but shortly afterwards widespread beyond the realm of random
matrices.

Starting with the celebrated Baik-Deift-Johansson Theorem [5], the
distribution $F_{\mathrm{GUE}}$ has been identified as the limiting one-point
distribution for the fluctuations of a wide range of different probabilistic
models. One of the most ubiquitous of such models is the KPZ equation,
introduced in the 1980s by Kardar, Parisi and Zhang. Despite numerous
developments surrounding it, exactly solving it remained an outstanding open
problem until the early 2010s, where four different groups of researchers [2,
57, 39, 26] independently found an exact solution for its so-called narrow
wedge solution. Amongst these works, Amir, Corwin and Quastel [2] found the
one-point distribution for the height function of the KPZ solution, showing
that it relates to a distribution found a little earlier by Johansson in a
grand canonical Gaussian-type matrix model [45], and further characterizing it
in terms of the integro-differential Painlevé II equation. The latter is an
extension of the PII differential equation, and almost as an immediate
consequence the authors of [2] also obtained that this one-point distribution,
in the large time limit, converges to $F_{\mathrm{GUE}}$ itself.

In much inspired by [5, 44] and later by [2, 57, 39, 26], it has been realized
that several stochastic growth models share an inherent connection with
statistics of integrable point processes, in what is formally established as
an identity between a transformation of the growth model and statistics for
the point process. To our knowledge, the very first instance of such relation
appears in the work of Borodin [15] which connects higher spin vertex model
with Macdonald measures. By taking appropriate limit of such connection, it
was later found that the KPZ equation is connected to the Airy2 point process
[18], ASEP is related to the discrete Laguerre Ensemble, stochastic six vertex
model is connected in the same way to the Meixner ensemble, or yet the
Krawtchouk ensemble [19].

As a common feature to these connections, the underlying correspondences
establish that the so-called $q$-Laplace transform of the associated height
function coincides with some multiplicative statistics of the point process.
The latter, in turn, admits exact solvability, and it is widely believed that
a long list of new insights on the growth models can be obtained by studying
the corresponding multiplicative statistics of the point processes.

This program has already been taken to a great start for the KPZ equation, and
exploring its connection with the Airy2 point process Corwin and Ghosal [30]
were able to obtain bounds for the lower tail of the KPZ equation. Shortly
afterwards, such bounds were improved by Cafasso and Claeys [24] with Riemann-
Hilbert methods common in random matrix theory.

Our major goal is to take on the program of understanding multiplicative
statistics for random particle systems, and carry out its detailed asymptotic
analysis for one of the most inspiring models, namely eigenvalues of random
matrices. While fluctuations of linear statistics of the eigenvalues of random
matrices has been extensively studied in the past, the study of multiplicative
statistics has only been carried over in particular instances, remarkably when
dealing with products and quotients of characteristic polynomials, see [38, 6,
1] and the references therein for a complete account.

In this work, we consider the Hermitian matrix model with an arbitrary one-cut
regular potential $V$, and associate to it a general family of multiplicative
statistics on its eigenvalues, indexed by a function $Q$ satisfying certain
regularity conditions. Our findings show that when the number of eigenvalues
is large such multiplicative statistics become universal: in the large matrix
limit they converge to a multiplicative statistics of the Airy2 point process
which is independent of $V$ and $Q$. This limiting statistics admits a
characterization in terms of a particular solution to the integro-differential
Painlevé II equation, and it is the same quantity that connects the KPZ
equation and the Airy2 point process. So, in turn, we find that random matrix
theory can recast the narrow wedge solution to KPZ equation for finite time in
an universal way.

The random matrix statistics that we study are associated to a deformed
orthogonal polynomial ensemble, also indexed by $Q$, which we analyze. As we
learn from earlier work of Borodin and Rains [20] which was recently
rediscovered and greatly extended by Claeys and Glesner [29] (and which we
also briefly explain later on), this deformed ensemble is a conditional
ensemble of a marked process associated to the original random matrix model.
We show that the correlation kernel for this point process converges to a
kernel constructed out of the same solution to the integro-differential PII
equation that appeared in [2]. This kernel is again universal in both $V$ and
$Q$, and turns out to be the kernel of the induced conditional process on the
marked Airy2 point process. Naturally, there are orthogonal polynomials and
their norming constants and recurrence coefficients associated to this
deformed ensemble. With our approach we also obtain similar universality
results for such quantities, showing that they are indeed universal in $V$ and
$Q$ and also connect to the integro-differential PII in a neat way.

Beyond the concrete results, with this work we also hope to shed light into
the rich structure underlying multiplicative statistics for eigenvalues of
random matrices. Much of the recent relevance of Painlevé equations is due to
its appearance in random matrix theory, see [40] for an overview of several of
these connections. There has been a growing recent interest in integro-
differential Painlevé-type equations [23, 25, 22, 46, 27, 51], and our results
place the integro-differential PII as a central universal object in random
matrix theory as well.

We scale the multiplicative statistics to produce a critical behavior at a
soft edge of the matrix model, and consequently the core of our asymptotic
analysis lies within the construction of a local approximation to all the
quantities near this critical point. Our main technical tool is the
application of the Deift-Zhou nonlinear steepest descent method to the
associated Riemann-Hilbert problem (shortly RHP), and the mentioned local
approximation is the so-called construction of a local parametrix. In our
case, a novel feat is that this local parametrix construction is performed in
a two-step way, first with the construction of a model problem varying with
large parameter, and second with the asymptotic analysis of this model
problem. In the latter, a RHP recently studied by Cafasso and Claeys [24] (see
also the subsequent works [25, 27]) which is related to the lower tail of the
KPZ equation shows up, and it is this RHP that ultimately connects all of our
considered quantities to the integro-differential PII.

The choice of scaling of our multiplicative statistics is natural,
illustrative but not exhaustive. As we point out later, with our approach it
becomes clear that other scalings could also be analyzed, say for instance
scaling around a bulk point, or yet soft/hard edge points with critical
potentials, and indicate that other integrable systems extending the integro-
differential PII may emerge.

##  2\. Statement of main results

Let
$\Lambda^{(n)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\lambda_{1}<\ldots<\lambda_{n})$
be a $n$-particle system with distribution

$\frac{1}{Z_{n}}\prod_{1\leq j<k\leq
n}(\lambda_{k}-\lambda_{j})^{2}\prod_{j=1}^{n}\operatorname{\rm
e}^{-nV(x)}\mathrm{d}x,$ (2.1)

where $Z_{n}$ is the partition function

$Z_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathbb{R}^{n}}\prod_{1\leq
j<k\leq n}(\lambda_{k}-\lambda_{j})^{2}\prod_{j=1}^{n}\operatorname{\rm
e}^{-nV(\lambda_{j})}\mathrm{d}\lambda_{1}\cdots\mathrm{d}\lambda_{n}.$ (2.2)

The distribution (2.1) is the eigenvalue distribution of the unitarily-
invariant random matrix model with potential function $V$ [33, 53].

We associate to $\Lambda^{(n)}$ the multiplicative statistics

$\displaystyle\mathsf{L}_{n}^{Q}(\mathsf{s})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\;\mathbb{E}\left(\prod_{j=1}^{n}\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{s}-n^{2/3}Q(\lambda_{j})}}\right)$ $\displaystyle=$
$\displaystyle\;\frac{1}{Z_{n}}\int_{\mathbb{R}^{n}}\prod_{1\leq j<k\leq
n}(\lambda_{k}-\lambda_{j})^{2}\prod_{j=1}^{n}\sigma_{n}(\lambda_{j})\operatorname{\rm
e}^{-nV(\lambda_{j})}\mathrm{d}\lambda_{1}\cdots\mathrm{d}\lambda_{n}=\frac{\mathsf{Z}^{Q}_{n}(\mathsf{s})}{Z_{n}},\quad\mathsf{s}>0,$
(2.3)

where $\mathsf{Z}_{n}^{Q}(\mathsf{s})$ is the partition function for the
deformed model

$\mathsf{Z}_{n}^{Q}(\mathsf{s})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathbb{R}^{n}}\prod_{1\leq
j<k\leq
n}(\lambda_{k}-\lambda_{j})^{2}\prod_{j=1}^{n}\sigma_{n}(\lambda_{j})\operatorname{\rm
e}^{-nV(\lambda_{j})}\mathrm{d}\lambda_{1}\cdots\mathrm{d}\lambda_{n}$ (2.4)

and we denoted

$\sigma_{n}(z)=\sigma_{n}(z\mid\mathsf{s})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(1+\operatorname{\rm
e}^{-\mathsf{s}-n^{2/3}Q(z)}\right)^{-1}.$

When $Q$ is linear, with a straightforward change of parameters
$\mathsf{L}^{Q}_{n}$ reduces to

$\mathbb{E}\left(\prod_{x\in\mathcal{X}}\frac{1}{1+\zeta q^{x}}\right),$ (2.5)

where the expectation is over the set $\mathcal{X}$ of configurations of
points (that is, for us $\mathcal{X}=\Lambda^{(n)}$) and $q\in(0,1)$ should be
viewed as a parameter of the model and, in general, $\zeta\in\mathbb{C}$ is a
free parameter. The expression (2.5) may be viewed as a transformation of the
point process, where $\zeta\in\mathbb{C}$ becomes the spectral variable of
this transformation, and the matching $\zeta=\operatorname{\rm
e}^{-\mathsf{s}}$ motivates the distinguished role of $\mathsf{s}$ in (2.3).
In the context of random particle systems, this particular multiplicative
statistics is associated to the notion of a $q$-Laplace transform [16, 18, 19,
17] that we already mentioned in the Introduction, and it has been one of the
key quantities in several outstanding recent progresses in asymptotics for
random particle systems [43, 30, 24].

We work under the following assumptions.

######  Assumptions 2.1.

  1. (i)

The potential $V$ is a nonconstant real polynomial of even degree and positive
leading coefficient, and its equilibrium measure $\mu_{V}$ is one-cut regular,
we refer to Section 8.1 below for the precise definitions. Performing a shift
on the variable, without loss of generality we assume that the right-most
endpoint of $\operatorname{supp}\mu_{V}$ is at the origin, so that

$\operatorname{supp}\mu_{V}=[-a,0],$

for some $a>0$.

  2. (ii)

The function $Q$ is real-valued over the real line, and analytic on a
neighborhood of the real axis. We also assume that it changes sign at the
right-most endpoint of $\operatorname{supp}\mu_{V}$, with

$Q(x)>0\text{ on }(-\infty,0),\quad Q(x)<0\text{ on }(0,\infty),$ (2.6)

with $x=0$ being a simple zero of $Q$. A particular role is played by the
negative value $Q^{\prime}(0)$, so we set

$\mathsf{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Q^{\prime}(0)>0.$
(2.7)

Although $\mathsf{t}$ in Assumption 2.1–(ii) will have the interpretation of
time, we stress that in this paper it will be kept fixed within a compact of
$(0,+\infty)$ rather than being made large or small.

For our results and throughout the whole work, we also talk about uniformity
of several error terms with respect to $\mathsf{t}$ in the sense that we now
explain. Because $Q$ is analytic on a neighborhood of the real axis, analytic
continuation shows that it is completely determined by its derivatives
$Q^{(k)}(0)$, $k\geq 0$. When we say that some error is uniform in
$\mathsf{t}$ within a certain range, we mean uniform when we vary $Q$ as a
function of $-Q^{\prime}(0)=\mathsf{t}$ while keeping all other derivatives
$Q^{(k)}(0)$, $k\geq 2$, fixed.

The condition in Assumption 2.1–(i) is standard in random matrix theory and
they are known to hold when, say, $V$ is a convex function [56]. The one-cut
assumption is made just for ease of presentation, as it simplifies the
Riemann-Hilbert analysis at the technical level considerably. On the other
hand, the regularity condition is used substantially in our arguments, but is
standard in Random Matrix Theory literature and holds true generically [49].
Most of our results are of local nature near the right-most endpoint of
$\operatorname{supp}\mu_{V}$ and could be shown to hold true for multi-cut
potentials near regular endpoints as well, with appropriate but non-essential
modifications.

Assumption 2.1–(ii) should be seen as specifying enough regularity on the
multiplicative statistics, here indexed by this factor $Q$. Because of
condition (ii), we have the pointwise convergence

$\sigma_{n}(x)\stackrel{{\scriptstyle
n\to\infty}}{{\to}}\begin{cases}0,&x>0,\\\ 1,&x<0,\end{cases}$ (2.8)

which means that the introduction of the factor $\sigma_{n}$ in the original
weight $\operatorname{\rm e}^{-nV}$ has the effect of producing an
interpolation between this original weight and its cut-off version
$\chi_{(-\infty,0)}\operatorname{\rm e}^{-nV}$, where from here onward
$\chi_{J}$ is the characteristic version of a set $J$. Comparing the Euler-
Lagrange conditions on the equilibrium problem induced by the weights
$\operatorname{\rm e}^{-nV}$ and $\chi_{(-\infty,0)}\operatorname{\rm
e}^{-nV}$, the observation we just made heuristically indicates that the
factor $\sigma_{n}$ does not change the global behavior of eigenvalues. This
may also be rigorously confirmed as an immediate consequence of our analysis,
but we do not elaborate on this end.

On the other hand, introducing a local coordinate $u=-z/n^{2/3}$ near the
origin, the approximation

$\operatorname{\rm e}^{-\mathsf{s}-n^{2/3}Q(z)}\approx\operatorname{\rm
e}^{-(\mathsf{t}u+\mathsf{s})}$

goes through, and we see that there is a competition between the term
$\mathsf{s}$ and $Q(z)$ that affects the local behavior of the weight at the
scale $\mathcal{O}(n^{-2/3})$ near the origin, which is the same scale for
nontrivial fluctuations of eigenvalues around the same point. The main results
that we are about to state concern obtaining the asymptotic behavior as
$n\to\infty$ of several quantities of the model, and in particular they
showcase how this term $Q$ affects the local scaling regime of the eigenvalues
near the origin and leads to connections with the integro-differential
Painlevé II equation as already mentioned.

A central object in this paper is the multiplicative statistics

$\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s,T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{E}\left(\prod_{j=1}^{\infty}\frac{1}{1+\operatorname{\rm
e}^{T^{1/3}(s+\mathfrak{a}_{j})}}\right),$ (2.9)

where the expectation is over the Airy2 point process with random
configuration of points $\\{\mathfrak{a}_{j}\\}$[55]. The quantity
$\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}$ admits two remarkable
characterizations, which are also of particular interest to us. The first is
the formulation via a Fredholm determinant, namely

$\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s,T)=\det\left(\mathbb{I}-\mathbb{K}^{{\rm
Ai}}_{T}\right)\raisebox{-2.15277pt}{$\big{|}$}_{L^{2}(-s,\infty)},$

where $\mathbb{K}^{{\rm Ai}}_{T}$ is the integral operator on
$L^{2}(-s,\infty)$ acting with the finite temperature (or fermi-type)
deformation of the Airy kernel $\mathsf{K}_{T}^{{\rm Ai}}$, defined by

$\mathsf{K}^{\rm{Ai}}_{T}(u,v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{-\infty}^{\infty}\frac{\operatorname{\rm
e}^{T^{1/3}\zeta}}{1+\operatorname{\rm
e}^{T^{1/3}\zeta}}\operatorname{Ai}(u+\zeta)\operatorname{Ai}(\zeta+v)\mathrm{d}\zeta,\quad
u,v\in\mathbb{R}.$

The term ‘temperature’ stems from the connection between the KPZ equation and
the random polymer models. Despite the name finite temperature, the parameter
$T$ here corresponds to the time in the KPZ equation, see (2.12) below. The
Fredholm determinant $\det\left(\mathbb{I}-\mathbb{K}^{\rm{Ai}}_{T}\right)$
appeared for the first time in the work of Johansson [45] as the limiting
process of a grand canonical (that is, when the number of particles/size of
matrix is also random) version of a Gaussian random matrix model, and
interpolates between the classical Airy kernel when $T\to+\infty$ and the
Gumbel distribution when $T\to 0^{+}$ with $s$ scaled appropriately. In [45,
Remark 1.13] Johansson already raises the question on whether a related
classical (that is, not grand canonical) matrix model has limiting local
statistics that interpolate between Gumbel and Tracy-Widom, as a similar
feature to $\det\left(\mathbb{I}-\mathbb{K}^{\rm{Ai}}_{T}\right)$. Since then,
other works have found $\det\left(\mathbb{I}-\mathbb{K}^{\rm{Ai}}_{T}\right)$
to be the limiting distribution for fluctuations around the largest particle
of a point process [32, 31, 51, 10]. In common, these works consider specific
models rather than obtaining
$\det\left(\mathbb{I}-\mathbb{K}^{\rm{Ai}}_{T}\right)$ as the universal limit
for a whole family of particle systems.

Another characterization of $\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}$ is
via a Tracy-Widom type formula that relates it to the integro-differential
PII. It reads

$\log\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(-\mathsf{S}\mathsf{T}^{1/3},\mathsf{T}^{-2})=-\frac{1}{\mathsf{T}}\int_{\mathsf{S}}^{\infty}(v-\mathsf{S})\left(\int_{-\infty}^{\infty}\Phi(r\mid
v,\mathsf{T})^{2}\frac{\operatorname{\rm e}^{-r}}{(1+\operatorname{\rm
e}^{-r})^{2}}\mathrm{d}r-\frac{v}{2}\right)\mathrm{d}v,$ (2.10)

where $\Phi$ solves the integro-differential Painlevé II equation

$\partial^{2}_{\mathsf{S}}\Phi(\xi\mid\mathsf{S},\mathsf{T})=\left(\xi+\frac{\mathsf{S}}{\mathsf{T}}+\frac{2}{\mathsf{T}}\int_{-\infty}^{\infty}\Phi(r\mid\mathsf{S},\mathsf{T})^{2}\frac{\operatorname{\rm
e}^{-r}}{(1+\operatorname{\rm
e}^{-r})^{2}}\mathrm{d}r\right)\Phi(\xi\mid\mathsf{S},\mathsf{T})$ (2.11)

with boundary value

$\Phi(\xi\mid\mathsf{S},\mathsf{T})\sim\mathsf{T}^{1/6}\operatorname{Ai}(\mathsf{T}^{2/3}\xi+\mathsf{S}\mathsf{T}^{-1/3}),\quad\text{as
}\xi\to\infty\text{ with }|\arg\xi|<\pi-\delta,$

for any $\delta>0$. This characterization has been obtained in the already
mentioned work by Amir, Corwin and Quastel [2], in connection with the narrow
wedge solution to the KPZ equation, and following the work [18] by Borodin and
Gorin has the interpretation that we now describe. For $\mathcal{H}(X,T)$
being the Hopf-Cole solution to the KPZ equation with narrow wedge initial
data at the space-time point $(X,T)$, introduce the rescaled random variable

$\Upsilon_{T}=\frac{\mathcal{H}(0,2T)+\frac{T}{12}}{T^{1/3}}.$

Based on the previous works [2, 26, 57, 39], in [18] the identity

$\mathbb{E}_{\mathrm{KPZ}}\left(\operatorname{\rm e}^{-\operatorname{\rm
e}^{T^{1/3}(\Upsilon_{T}+s)}}\right)=\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s,T)$
(2.12)

between the height function of the KPZ equation and the multiplicative
statistics $\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s,T)$ is identified.
This is an instance of matching formulas relating growth processes with
determinantal point processes that we already mentioned at the Introduction.
One of the key aspects of this representation is that the Airy2 point process
is determinantal, and consequently its statistics can be studied using
techniques from exactly solvable/integrable models. Indeed Equation 2.12 is
the starting point taken by Cafasso and Claeys [24], who then connected
$\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s,T)$ to a RHP that will also
play a major role for us. Recently, Cafasso, Claeys and Ruzza [25] also
obtained an independent proof of the representation (2.10), extending it to
more general multiplicative statistics of the Airy2 point process. Other
proofs and extensions of this integro-differential equation have also been
recently found in related contexts [23, 46, 22]. Also, by exploring (2.12) the
tail behavior of the KPZ equation has become rigorously accessible in various
asymptotic regimes [30, 24, 25, 27].

As a first result, we prove that the multiplicative statistics
$\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}$ is the universal limit of
$\mathsf{L}_{n}^{Q}(\mathsf{s})$.

######  Theorem 2.2.

Suppose that $V$ and $Q$ satisfy Assumptions 2.1 and fix $\mathsf{s}_{0}>0$
and $\mathsf{t}_{0}\in(0,1)$. For a constant $\mathsf{c}_{V}>0$ that depends
solely on $V$, and any $\nu\in(0,2/3)$, the asymptotic estimate

$\log\mathsf{L}^{Q}_{n}\left(\mathsf{s}\right)=\log\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}\left(-\frac{\mathsf{s}\mathsf{c}_{V}}{\mathsf{t}},\frac{\mathsf{t}^{3}}{\mathsf{c}_{V}^{3}}\right)+\mathcal{O}(n^{-\nu}),\quad
n\to\infty$ (2.13)

holds true uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$.

Findings on random matrix theory surrounding the Tracy-Widom distribution have
inspired an enormous development in the KPZ universality theory. For the KPZ
equation one of the major developments can be phrased by saying that the
fluctuations of the height function for the narrow wedge solution coincide, in
the large time limit, with the $\beta=2$ Tracy-Widom law from random matrix
theory. Theorem 2.2 is saying that the connection between random matrix theory
and the KPZ equation can be recast already at any finite time, and not only
for Gaussian models but also universally in $V$ and $Q$. Similar connection
exists [8] between the solution of the KPZ equation in half-space under the
Robin boundary condition and Airy1 point process which, in turn, in the large
time limit relate this KPZ solution to GOE matrices.

The constant $\mathsf{c}_{V}$ is determined from (8.2) and (8.5) below. It is
the first derivative of a conformal map near the origin, which is constructed
out of the equilibrium measure for $V$. Ultimately, we make a conformal change
of variables of the form $\zeta\approx\mathsf{c}_{V}z/n^{2/3}$, which in turn
identifies

$\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{s}-n^{2/3}Q(z)}}\approx\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta/\mathsf{c}_{V}}}.$

In light of (2.9), this explains the evaluation
$s=-\mathsf{s}\mathsf{c}_{V}/\mathsf{t}$ and
$T=\mathsf{t}^{3}/\mathsf{c}_{V}^{3}$ on the right-hand side of (2.13).

We emphasize that the error term in (2.13) is not in sharp form. In Section
3.1 we explain how this term arises from our techniques. We do not have
indications regarding whether the true optimal error would be
$\mathcal{O}(n^{-2/3})$ (or of any polynomial order) or if it should involve,
say, logarithmic corrections.

Our next results concern limiting asymptotic formulas for the matrix model
underlying $\mathsf{L}_{n}^{Q}$, starting with the partition function
$\mathsf{Z}_{n}^{Q}$ from (2.2). For a polynomial $V$, its deformation

$V_{t}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(1-\frac{1}{t}\right)z^{2}+V(z/t),\quad
1\leq t\leq+\infty$ (2.14)

has the property that $V_{1}=V$ and $V_{\infty}=z^{2}$ is Gaussian. Under the
assumption that $V_{t}$ is one-cut regular for every $t\geq 1$, Bleher and Its
[14] proved that an expansion of the form

$Z_{n}=Z_{n}^{\mathrm{GUE}}\operatorname{\rm
e}^{n^{2}\mathbf{e}_{0}^{V}+\mathbf{e}_{1}^{V}}\left(1+\mathcal{O}(n^{-2})\right),\quad
n\to\infty,$ (2.15)

holds, where $Z_{n}^{\mathrm{GUE}}$ is the GUE partition function for
$V_{\infty}=z^{2}$ and $\mathbf{e}_{0}^{V}$ and $\mathbf{e}_{1}^{V}$ are
functions analytic on the coefficients of $V$. In fact, their result ensures a
full asymptotic expansion in inverse powers of $n^{2}$, see also [9, 41, 13]
for important earlier work obtaining similar results under different
conditions, and also the more recent contributions [12, 21].

As an immediate corollary to Theorem 2.2 we obtain some terms in the
asymptotic expansion of the deformed partition function (2.4).

######  Corollary 2.3.

Suppose $V$ and $Q$ satisfy Assumptions 2.1 and fix $\mathsf{s}_{0}>0$ and
$\mathsf{t}_{0}>0$. In addition, assume that the potential $V_{t}$ in (2.14)
is one-cut regular for any $t\geq 1$. For any $\nu\in(0,2/3)$, the partition
function admits an expansion of the form

$\mathsf{Z}_{n}^{Q}(\mathsf{s})=Z_{n}^{\mathrm{GUE}}\operatorname{\rm
e}^{n^{2}\mathbf{e}_{1}^{V}+\mathbf{e}_{0}^{V}}\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}\left(-\frac{\mathsf{s}\mathsf{c}_{V}}{\mathsf{t}},\frac{\mathsf{t}^{3}}{\mathsf{c}_{V}^{3}}\right)\left(1+\mathcal{O}(n^{-\nu})\right),\quad
n\to\infty$ (2.16)

which is valid uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$, and the coefficients
$\mathbf{e}_{0}^{V}$ and $\mathbf{e}_{1}^{V}$ are as in (2.15).

###### Proof.

Follows from the expansion (2.15), Theorem 2.2 and (2.3). ∎

The order of error (2.16) is not $n^{-2}$ as in (2.15) but weaker and not
sharp. This phenomenon can be traced back to the fact that $\sigma_{n}$ has
infinitely many poles accumulating on the real axis as $n\to\infty$, see the
discussion in Section 3.1 below. A similar error order was obtained in [14,
Theorem 9.1 and Equation (9.68)], in a transitional regime from a one-cut to
two-cut potential, and where the role played here by
$\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}$ is replaced by the GUE Tracy-
Widom distribution itself.

From the general theory of unitarily invariant random matrix models, it is
known that the density appearing in (2.3) admits a determinantal form. Setting

$\omega_{n}(z)=\omega^{Q}_{n}(z\mid\mathsf{s})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sigma_{n}(z)\operatorname{\rm
e}^{-nV(z)},\quad\text{where we
recall}\quad\sigma_{n}(z)=\left(1+\operatorname{\rm
e}^{-\mathsf{s}-n^{2/3}Q(z)}\right)^{-1},$ (2.17)

this means that the identity

$\frac{1}{\mathsf{Z}_{n}(\mathsf{s})}\prod_{1\leq j<k\leq
n}(\lambda_{k}-\lambda_{j})^{2}\prod_{j=1}^{n}\sigma_{n}(\lambda_{j})\operatorname{\rm
e}^{-nV(\lambda_{j})}=\frac{1}{n!}\det\left(\omega_{n}(\lambda_{j})^{1/2}\mathsf{K}^{Q}_{n}(\lambda_{j},\lambda_{k})\omega_{n}(\lambda_{k})^{1/2}\right)_{j,k=1}^{n}$

holds true for a function of two variables $\mathsf{K}^{Q}_{n}(x,y)$
satisfying certain properties, known as the correlation kernel of the
eigenvalue density on the left-hand side. The correlation kernel is not
unique, but in the present setup it may be taken to be the Christoffel-Darboux
kernel for the orthogonal polynomials for the weight $\omega_{n}$, as we
introduce in detail in (9.3), and whenever we talk about $\mathsf{K}_{n}^{Q}$
we mean this Christoffel-Darboux kernel. In particular,
$\mathsf{K}^{Q}_{n}=\mathsf{K}^{Q}_{n}(\cdot\mid\mathsf{s})$ does depend on
both $Q$ and $\mathsf{s}$.

Our second result proves universality of the kernel $\mathsf{K}_{n}^{Q}$,
showing that its limit depends solely on $\mathsf{s}$ and
$\mathsf{t}=-Q^{\prime}(0)$, but not on other aspects on $Q$, and relates to
the integro-differential PII. For its statement, it is convenient to introduce
the new set of variables

$\mathsf{T}=\mathsf{t}^{-3/2}\quad\text{and}\quad\mathsf{S}=\mathsf{s}\mathsf{t}^{-3/2}.$
(2.18)

With $\Phi(\xi)=\Phi(\xi\mid\mathsf{S},\mathsf{T})$ being the solution to the
integro-differential Painlevé II equation in (2.11) and the variables
$\mathsf{s},\mathsf{t}$ and $\mathsf{S},\mathsf{T}$ related by (2.18), we set

$\upphi_{1}(\zeta\mid\mathsf{s},\mathsf{t})=\Phi(\xi(\zeta)\mid\mathsf{S},\mathsf{T}),\quad\upphi_{2}(\zeta\mid\mathsf{s},\mathsf{t})=(\partial_{\mathsf{S}}\Phi)(\xi(\zeta)\mid\mathsf{S},\mathsf{T}),\quad\quad\xi(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\mathsf{s}+\mathsf{t}\zeta,$

and introduce the kernel

$\mathsf{K}_{\infty}(u,v\mid\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\upphi_{1}(v\mid\mathsf{s},\mathsf{t})\upphi_{2}(u\mid\mathsf{s},\mathsf{t})-\upphi_{1}(u\mid\mathsf{s},\mathsf{t})\upphi_{2}(v\mid\mathsf{s},\mathsf{t})}{u-v},\quad
u,v\in\mathbb{R}.$

######  Theorem 2.4.

Assume that $V$ and $Q$ satisfy Assumptions 2.1 and fix $\mathsf{s}_{0}>0$ and
$\mathsf{t}_{0}\in(0,1)$. With

$u_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{u}{\mathsf{c}_{V}n^{2/3}},\quad
v_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{v}{\mathsf{c}_{V}n^{2/3}},$
(2.19)

the estimate

$\frac{\operatorname{\rm
e}^{-\frac{n}{2}(V(u_{n})+V(v_{n}))}}{\mathsf{c}_{V}n^{2/3}}\mathsf{K}^{Q}_{n}\left(u_{n},v_{n}\mid\mathsf{s}\right)=\mathsf{K}_{\infty}(u,v\mid\mathsf{s},\mathsf{t}/\mathsf{c}_{V})+\mathcal{O}(n^{-1/3}),\quad
n\to\infty,$ (2.20)

holds true uniformly for $u,v$ in compacts of $\mathbb{R}$, and uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$.

In the recent work [29], Claeys and Glesner developed a general framework for
certain conditional point processes, which in particular yields a
probabilistic interpretation of the kernel $\mathsf{K}_{n}^{Q}$ as we now
explain. For a point process $\Lambda$, we add a mark $0$ to a point
$\lambda\in\Lambda$ with probability $\sigma_{n}(\lambda)$ and a mark $1$ with
complementary probability $1-\sigma_{n}(\lambda)$. This induces a
decomposition of the point process $\Lambda=\Lambda_{0}\cup\Lambda_{1}$, where
$\Lambda_{j}$ is the set of eigenvalues with mark $j$. We then consider the
induced point process $\widehat{\Lambda}$ obtained from $\Lambda$ upon
conditioning that $\Lambda_{1}=\emptyset$, that is, that all points have mark
$0$.

When applied to the eigenvalue point process $\Lambda=\Lambda^{(n)}$ induced
by the distribution (2.1), the theory developed in [29] shows that
$\widehat{\Lambda}^{(n)}$ is a determinantal point process with correlation
kernel proportional to
$\omega_{n}(x)^{1/2}\omega_{n}(y)^{1/2}\mathsf{K}_{n}^{Q}(x,y)$ which, in
turn, generates the same point process as the left-hand side of (2.20), see
[29, Sections 4 and 5]. A comparison of the RHP that characterizes the kernel
$\mathsf{K}_{\infty}$ (see Section 5.1 below, in particular (5.16)) with the
discussion in [29, Section 5.2] shows that $\mathsf{K}_{\infty}$ is a
(renormalized) correlation kernel for the marked point process
$\widehat{\\{\mathfrak{a}_{k}\\}}$ of the Airy2 point process
$\\{\mathfrak{a}_{k}\\}$ with the marking function $(1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\lambda})^{-1}$. So Theorem 2.4 assures that the
conditional process on the marked eigenvalues converges, at the level of
rescaled correlation kernels, to the conditional process on the marked Airy2
point process.

We also obtain asymptotics for the norming constant
$\upgamma^{(n,Q)}_{n-1}(\mathsf{s})$ for the ${(n-1)}$-th monic orthogonal
polynomial for the weight $\omega_{n}(x\mid\mathsf{s})$ (see (9.2) for the
definition), showing that its first correction term depends again solely on
$\mathsf{s},\mathsf{t}$, and also relates to the integro-differential Painlevé
II equation.

######  Theorem 2.5.

Suppose that $V$ and $Q$ satisfy Assumptions 2.1 and fix $\mathsf{s}_{0}>0$
and $\mathsf{t}_{0}\in(0,1)$. The norming constant has asymptotic behavior

$\upgamma^{(n,Q)}_{n-1}(\mathsf{s})^{2}=\frac{a}{4\pi}\operatorname{\rm
e}^{-2n\ell_{V}}\left(\frac{1}{2}-\frac{1}{n^{1/3}}\frac{\mathsf{c}_{V}^{1/2}}{\mathsf{t}^{1/2}}\left(\mathsf{p}(\mathsf{s},\mathsf{t}/\mathsf{c}_{V})-\frac{\mathsf{s}^{2}\mathsf{c}_{V}^{3/2}}{4\mathsf{t}^{3/2}}\right)+\mathcal{O}(n^{-2/3})\right),\quad
n\to\infty,$ (2.21)

uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$, where
$\mathsf{p}(\mathsf{s},\mathsf{t})=\mathsf{P}(\mathsf{S},\mathsf{T})$, and the
function $\mathsf{P}$ relates to the solution $\Phi$ from (2.11) via

$\partial_{\mathsf{S}}\mathsf{P}(\mathsf{S},\mathsf{T})=\frac{\mathsf{S}}{2\mathsf{T}}+\frac{1}{\mathsf{T}}\int_{-\infty}^{\infty}\Phi(r\mid\mathsf{S},\mathsf{T})^{2}\frac{\operatorname{\rm
e}^{-r}}{(1+\operatorname{\rm e}^{-r})^{2}}\mathrm{d}r.$ (2.22)

Our approach also yields asymptotic formulas for the orthogonal polynomials
and their recurrence coefficients, and relate them to the integro-differential
Painlevé II equation as well, but for the sake of brevity we do not state
them.

##  3\. About our approach: issues and extensions

###  3.1. Issues to be overcome

Our main tool for obtaining all of our results is the Fokas-Its-Kitaev [42]
Riemann-Hilbert Problem (RHP) for orthogonal polynomials (shortly OPs) that
encodes the correlation kernel $\mathsf{K}_{n}^{Q}$, the norming constants
$\upgamma^{(n,Q)}_{n-1}(\mathsf{s})^{2}$ and ultimately also the
multiplicative statistics $\mathsf{L}_{n}^{Q}$, and its asymptotic analysis
via the Deift-Zhou nonlinear steepest descent method [35, 37]. The overall
arch of this asymptotic analysis is the usual one, summarized in the diagram
in Figure 1, and we now comment on its major steps.

pt$\mathbf{Y}$ptRHP for OPsptintroduction of
$g$-functionptpt$\mathbf{T}$ptopening of lensesptpt$\mathbf{S}$ptlocal
parametrix near
$z=0$$\mathbf{P}(z)=\mathbf{E}_{n}(z)\mathbf{\Phi}_{n}(n^{2/3}\psi(z))$ptptglobal
parametrix$\mathbf{G}(z)$ptptindependent
step$\mathbf{\Phi}_{n}(\zeta)\to{\mathbf{\Phi}}_{0}(\zeta)$pt$\mathbf{R}$
Figure 1. Schematic diagram for the steps in the asymptotic analysis of the
RHP for orthogonal polynomials. There is also an Airy local parametrix used
near $z=-a$ which we omit in this diagram.

Starting with the RHP for OPs that we name $\mathbf{Y}$, in the first step we
transform $\mathbf{Y}\mapsto\mathbf{T}$ with the introduction of the
$g$-function (or, equivalently, the $\phi$-function), and this is done so with
the help of the equilibrium measure for $V$ that accounts only for the part
$\operatorname{\rm e}^{-nV}$ of the weight $\omega_{n}$. In the second step,
we open lenses with a transformation $\mathbf{T}\mapsto\mathbf{S}$ as usual.

The third step is the construction of the global parametrix $\mathbf{G}$. In
our case, in this construction we also have to account for the perturbation
$\sigma_{n}$ of the weight $\operatorname{\rm e}^{-nV}$, so a Szegö function-
type construction is used.

The fourth step is the construction of local parametrices at the endpoints
$z=-a,0$ of $\operatorname{supp}\mu_{V}$, with the goal of approximating all
the quantities locally near these endpoints. This is accomplished by, first,
considering a change of variables $z\mapsto\zeta=n^{2/3}\psi(z)$ after the
conformal map $\psi$ chosen appropriately for each endpoint and, then,
constructing the solution to a model RHP $\mathbf{\Phi}(\zeta)$ in the
$\zeta$-plane. Following these steps, the local parametrix at the left edge
$z=-a$ of $\operatorname{supp}\mu_{V}$ is standard and utilizes Airy
functions.

The construction of the local parametrix at the right edge $z=0$ is, however,
a lot more involved. As we mentioned earlier, the factor $\sigma_{n}$ affects
asymptotics of local statistics near the origin. In fact, the weight
$\sigma_{n}$ has singularities precisely at the points of the form

$-\frac{\mathsf{s}}{n^{2/3}}+\frac{\pi\mathrm{i}(2k+1)}{n^{2/3}},\quad
k\in\mathbb{Z}.$

This means that for $\mathsf{s}\ll n^{2/3}$ there are infinitely many poles of
$\sigma_{n}$ accumulating near the real axis. As such, in this case for large
$n$ the perturbed weight $\omega_{n}$ fails to be analytic in any fixed
neighborhood of the origin. If we were to consider only $\mathsf{s}\to+\infty$
fast enough, one could still push the standard RHP analysis further with the
aid of Airy local parametrices, at the cost of a worse error estimate.
However, when $\mathsf{s}=\mathcal{O}(1)$ we have poles accumulating too fast
to the real axis, and a different asymptotic analysis has to be accomplished,
in particular a new local parametrix is needed.

When changing coordinates $z\mapsto\zeta$ near $z=0$, the model problem
$\mathbf{\Phi}=\mathbf{\Phi}_{n}$ obtained is then $n$-dependent. This is so
because the jump of the local parametrix involves $\sigma_{n}$, and
consequently in the process of changing variables the resulting model problem
has a jump that involves a transformation of $\sigma_{n}$ itself. This is in
contrast with usual constructions with, say, Airy, Bessel or Painlevé-type
parametrices, where the jumps can be turned into piecewise constant in the
$z$-plane and, hence, also remain piecewise constant in the $\zeta$-plane.
Another feature of the RHP for the model problem $\mathbf{\Phi}_{n}$ is that
its jump is not analytic on the whole plane, and instead it is analytic only
in a growing (with $n$) disk, and for a fixed $n$ we can only ensure that its
jump matrix is $C^{\infty}$ in a neighborhood of the jump contour.

All in all, this means that carrying out the asymptotic analysis of
$\mathbf{\Phi}_{n}(\zeta)$ as $n\to\infty$ is also needed. As we said, the
jump for $\mathbf{\Phi}_{n}$ involves a transformation of $\sigma_{n}$, so
ultimately also depends on the function $Q$ from (2.17). But it turns out that
as $n\to\infty$, we have the convergence
$\mathbf{\Phi}_{n}\to\mathbf{\Phi}_{0}$ in an appropriate sense, where
$\mathbf{\Phi}_{0}$ is independent of $Q$. This limiting $\mathbf{\Phi}_{0}$
is the solution to a RHP that appeared recently in connection with the KPZ
equation [24] and which was later shown to connect with the integro-
differential PII in the recent work of Claeys, Cafasso and Ruzza [25, 27]. For
this reason we term it the id-PII RHP.

With the construction of the global and local parametrices, the asymptotic
analysis is concluded in the usual way, by patching them together and
obtaining a new RHP for a matrix function $\mathbf{R}$. This matrix
$\mathbf{R}$, in turn, solves a RHP whose jump is asymptotically close to the
identity, and consequently $\mathbf{R}$ can be found perturbatively.

After concluding this asymptotic analysis, we undress the transformations
$\mathbf{R}\mapsto\cdots\mapsto\mathbf{Y}$ and obtain asymptotic expressions
for the wanted quantities. For the kernel $\mathsf{K}_{n}^{Q}$ and the norming
constant $\upgamma^{(n,Q)}_{n-1}(\mathsf{s})^{2}$, after this undressing
Theorems 2.4 and 2.5 follow in a standard manner.

However, to obtain (2.13) quite some extra work is needed. When dealing with
statistics of matrix models via OPs, one of the usual approaches is to extract
the needed information via the partition function and its relation with the
norming constants via a product formula, see for instance (9.6) below. Usually
this is accomplished via some differential identity or with careful estimate
of each term in the product formula, see for instance the works [12, 14, 4,
47] and their references for explorations along these lines. In virtue of the
relation (2.3) this was in fact our original attempt, but several technical
issues arise. Instead, at the end we express $\mathsf{L}_{n}^{Q}$ directly as
a weighted double integral of $\mathsf{K}_{n}^{Q}(x,x\mid\mathsf{s})$ in the
variables in $x$ and $\mathsf{s}$, this is done in Proposition 9.1 below. The
$x$-integral takes place over the whole real line, which means that when we
undress $\mathbf{R}\mapsto\mathbf{Y}$ we obtain a formula for
$\mathsf{L}_{n}^{Q}$ involving global and all local parametrices. The integral
in $\mathsf{s}$ extends to $+\infty$, which is one of the main reasons why in
our main statements we also keep track of uniformity of errors when
$\mathsf{s}\to+\infty$. We then have to estimate the double integral,
accounting for exponential decays of most of the terms but also exact
cancellations of some other terms. Ultimately, the whole analysis leads to a
leading contribution coming solely from a portion of the integral that arises
from the model problem $\mathbf{\Phi}_{n}$. With a further asymptotic analysis
of the later integral we obtain an integral solely of ${\mathbf{\Phi}}_{0}$
which then yields Theorem 2.2.

The convergence $\mathbf{\Phi}_{n}\to{\mathbf{\Phi}}_{0}$ is treated as a
separate issue, and to achieve it we need several information about this id-
PII parametrix ${\mathbf{\Phi}}_{0}$. As a final outcome, we obtain that
$\mathbf{\Phi}_{n}$ is close to ${\mathbf{\Phi}}_{0}$ with an error term of
the form $\mathcal{O}(n^{-\nu})$, for any $\nu\in(0,2/3)$. But, in much due to
the non-analyticity of the jump matrix for $\mathbf{\Phi}_{n}$, we are not
able to achieve a sharp order $\mathcal{O}(n^{-2/3})$ unless further
conditions were placed on $Q$. This non-optimal error explains the appearance
of the same error order in (2.13). In the course of this asymptotic analysis
we rely substantially in [25]. Among other needed info, we also borrow from
the same work the connection of ${\mathbf{\Phi}}_{0}$ with the integro-
differential PII. In the same work, the authors actually show that
${\mathbf{\Phi}}_{0}$ relates to particular solutions to the KdV equation that
reduce to the integro-differential PII. As such, Theorems 2.4 and 2.5 could be
phrased in terms of a solution to the KdV rather than to the integro-
differential PII. We opt to phrase them with the latter because this
formulation encodes that all self-similarities have already been accounted
for.

If we were to assume that the jump matrix for $\mathbf{\Phi}_{n}$ were
piecewise analytic on the whole plane and not merely $C^{\infty}$, we could
deform $\mathbf{\Phi}_{n}$ to a family of RHPs considered in [25]. With this
in mind, the analysis of the convergence
$\mathbf{\Phi}_{n}\to{\mathbf{\Phi}}_{0}$ is inspired by several aspects in
this just mentioned work but, as we already said, here we are forced to work
under different conditions on the jump matrix. In particular, one could adapt
the methods in [25] to actually prove that $\mathbf{\Phi}_{n}$ does too relate
to an $n$-dependent solution to the integro-differential PII. Consequently,
with a careful inspection of our work one could show that Theorems 2.2, 2.4
and 2.5 admit versions with $n$-dependent leading terms. For instance,
relating the norming constant $\upgamma^{(n,Q)}_{n-1}(\mathsf{s})^{2}$ with
the model problem $\mathbf{\Phi}_{n}$ one could obtain an asymptotic formula
of the form

$\upgamma^{(n,Q)}_{n-1}(\mathsf{s})^{2}=\frac{a}{4\pi}\operatorname{\rm
e}^{-2n\ell_{V}}\left(\frac{1}{2}-\frac{1}{n^{1/3}}\frac{\mathsf{c}_{V}^{1/2}}{\mathsf{t}^{1/2}}\left(\mathsf{p}_{n}(\mathsf{s},\mathsf{t}/\mathsf{c}_{V})-\frac{\mathsf{s}^{2}\mathsf{c}_{V}^{3/2}}{4\mathsf{t}^{3/2}}\right)+\mathcal{O}(n^{-2/3})\right),\quad
n\to\infty,$

where the $n$-dependent function $\mathsf{p}_{n}$ is obtained from
$\mathbf{\Phi}_{n}$ and relates to a $n$-dependent solution $\Phi_{n}$ to the
integro-differential PII. In fact, with standard arguments one could improve
the formula above to a full asymptotic expansion in powers of $n^{-1/3}$, with
bounded but $n$-dependent coefficients. Underlying our arguments there is the
statement that $\mathsf{p}_{n}=\mathsf{p}+\mathcal{O}(n^{-\nu})$ for any
$\nu\in(0,2/3)$, which then yields Theorem 2.5. But as a drawback, although
one could potentially improve (2.21) and also obtain the term of order
$n^{-2/3}$ explicitly, it is not possible to obtain the $\mathcal{O}(n^{-1})$
term in (2.21) unless one improves the error $\mathcal{O}(n^{-\nu})$ in the
convergence $\mathbf{\Phi}_{n}\to{\mathbf{\Phi}}_{0}$ to a sharp error
$\mathcal{O}(n^{-2/3})$.

###  3.2. Possible extensions

Most of our approach may be extended to potentials $V$ for which the
equilibrium measure $\mu_{V}$ is critical, and also under different conditions
on $Q$ as we now explain.

Apart from technical adaptations in several steps of the RHP for OPs which are
nowadays well understood, our analysis carries over to potentials $V$ for
which the equilibrium measure $\mu_{V}$ is regular but multicut, with the same
conditions on $Q$ when $\mu_{V}$ has the origin as its right-most endpoint.

When, say, the density $\mu_{V}$ vanishes to a higher power at a soft edge
and/or $Q$ changes sign with an arbitrary odd vanishing order at the same soft
edge, we need to replace the power $n^{2/3}$ in $\sigma_{n}$ by another
appropriate power to modify the local statistics near this point in a non-
trivial critical manner. Once this is done, the asymptotic analysis of the RHP
for OPs that we perform carries over mostly with minor modifications, and the
only major issue to overcome is in the construction of a new local parametrix
$\widetilde{\mathbf{\Phi}}_{n}$ near this soft edge point and its
corresponding asymptotic analysis. In this case, we expect that
$\widetilde{\mathbf{\Phi}}_{n}\to\widetilde{\mathbf{\Phi}}_{0}$ for a new
function $\widetilde{\mathbf{\Phi}}_{0}$. It is relatively simple to write a
RHP that should be satisfied by this $\widetilde{\mathbf{\Phi}}_{0}$, and we
expect it to be related to the KdV hierarchy [28] but with nonstandard initial
data. It would be interesting to see if the particular solutions obtained this
way reduce to integro-differential hierarchies of Painlevé equations, in the
same spirit of the recent works [23, 46].

One could also consider similar statistics to (2.3) with a $Q$ that vanishes
at a bulk point of $\operatorname{supp}\mu_{V}$. We do expect that most of our
work carries through to this situation, at least when we impose $V$ to be
again one-cut regular and $Q$ to vanish quadratically at a point inside
$\operatorname{supp}\mu_{V}$. The main issue that should arise is again on the
construction of the local parametrix near this point, and its corresponding
asymptotic analysis. This model should lead to multiplicative statistics of
the Sine kernel (and the higher order generalizations of it). Similar
considerations go through to hard-edge models, leading to multiplicative
statistics of the Bessel process. To our knowledge, such multiplicative
statistics of Bessel and Sine have not been considered in the literature so
far. However, finite temperature versions of the Sine and Bessel kernels do
have appeared, see for instance [45, 10, 11, 31].

###  3.3. Organization of the paper

The paper is organized in two parts. In the first part, we deal with a family
of RHPs $\mathbf{\Phi}_{\tau}$ that contains the model RHP $\mathbf{\Phi}_{n}$
needed in the asymptotic analysis of OPs. In Section 4 we introduce
$\mathbf{\Phi}_{\tau}$ formally. In Sections 5 and 6 we discuss the RHP
${\mathbf{\Phi}}_{0}$, which is a particular case of $\mathbf{\Phi}_{\tau}$,
and review several of its properties, translating results from [24, 25] to our
notation and needs. In Section 7 we prove the convergence
$\mathbf{\Phi}_{\tau}\to{\mathbf{\Phi}}_{0}$ and of related quantities in the
appropriate sense. The latter section contains all the needed results for the
asymptotic analysis of the RHP for OPs, and concludes the first part of this
paper.

The second part of the paper is focused on the asymptotic analysis of the RHP
for OPs. In Section 8 we discuss several aspects that relate to the
equilibrium measure. In Section 9 we introduce the Christoffel-Darboux kernel
$\mathsf{K}_{n}^{Q}$ and related quantities, and display how they relate to
the RHP for OPs. In particular, in Proposition 9.1 we write
$\mathsf{L}_{n}^{Q}$ directly to as an integral of the kernel
$\mathsf{K}_{n}^{Q}$, a result which may be of independent interest. In
Section 10 we perform the asymptotic analysis of the RHP for the OPs. In
Section 11 use the conclusions from Sections 10 and 7 and prove Theorems 2.4
and 2.5. Also from the results from Sections 10 and 7 and assuming additional
technical estimates, the proof of Theorem 2.2 is given in Section 11. Such
remaining technical estimates are also ultimately a consequence of the RHP
analysis, but their proofs are rather cumbersome and postponed to Section 12.

For the remainder of the paper it is convenient to denote

$\mathbf{e}_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{pmatrix}1\\\
0\end{pmatrix},\quad\mathbf{e}_{2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{pmatrix}0\\\
1\end{pmatrix},\quad\mathbf{E}_{jk}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{e}_{j}\mathbf{e}_{k}^{\mathrm{T}},$
(3.1)

so $\mathbf{E}_{jk}$ is a $2\times 2$ matrix with the $(j,k)$-entry equals $1$
and all other entries zero. With this notation, the Pauli matrices, for
instance, take the form

$\mathbf{I}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{E}_{11}+\mathbf{E}_{22},\quad\bm{\sigma}_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{E}_{12}+\mathbf{E}_{21},\quad\bm{\sigma}_{2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\mathrm{i}\mathbf{E}_{12}+\mathrm{i}\mathbf{E}_{21},\quad\bm{\sigma}_{3}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{E}_{11}-\mathbf{E}_{22}.$
(3.2)

In particular, for any reasonably regular scalar function $f$, the spectral
calculus yields

$f(z)^{\bm{\sigma}_{3}}=\begin{pmatrix}f(z)&0\\\ 0&1/f(z)\end{pmatrix}.$ (3.3)

These notations will be used extensively in the coming sections.

### Acknowledgments

P. G. wishes to thank Ivan Corwin and Alexandre Krajenbrink for many helpful
conversations and Alexei Borodin for comments on the earlier version of this
manuscript. G. S. is grateful to Jinho Baik, Tom Claeys, Mattia Cafasso, Lun
Zhang and Alfredo Deaño for inspiring conversations, and Dan Betea for
pointing us out to relevant references. He also acknowledges his current
support by São Paulo Research Foundation under grants # 2019/16062-1 and #
2020/02506-2, and by Brazilian National Council for Scientific and
Technological Development (CNPq) under grant # 315256/2020-6. This work was
partially developed while the authors participated in the program Universality
and Integrability in Random Matrix Theory and Interacting Particle Systems,
hosted by the Mathematical Sciences Research Institute in Berkeley,
California, during the Fall 2021 semester, supported by the National Science
Foundation under Grant No. DMS-1928930. We are grateful to the organizers for
their effort in providing an excellent research atmosphere despite the
uncertain times.

##  4\. A Model Riemann-Hilbert Problem

In this section we discuss a model Riemann-Hilbert Problem that will be used
in the construction of a local parametrix in the asymptotic analysis for the
orthogonal polynomials. As such, this model problem plays a central role in
obtaining all our major results.

###  4.1. The model problem

Set

$\mathsf{\Sigma}_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[0,+\infty),\;\mathsf{\Sigma}_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[0,\operatorname{\rm
e}^{2\pi\mathrm{i}/3}),\;\mathsf{\Sigma}_{2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(-\infty,0],\;\mathsf{\Sigma}_{3}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[0,\operatorname{\rm
e}^{-2\pi\mathrm{i}/3}),\quad\mathsf{\Sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\bigcup_{j=0}^{3}\mathsf{\Sigma}_{j},$
(4.1)

orienting $\mathsf{\Sigma}_{0}$ from the origin to $\infty$, and the remaining
arcs from $\infty$ to the origin, see Figure 2.

$\mathbb{R}$$\mathrm{i}\mathbb{R}$$\mathsf{\Sigma}_{0}$$\mathsf{\Sigma}_{1}$$\mathsf{\Sigma}_{2}$$\mathsf{\Sigma}_{3}$$\scriptstyle\frac{2\pi}{3}$
Figure 2. The contours
$\mathsf{\Sigma}_{0},\mathsf{\Sigma}_{1},\mathsf{\Sigma}_{2}$ and
$\mathsf{\Sigma}_{3}$ in (4.1) that constitute $\mathsf{\Sigma}$.

The model RHP we are about to introduce depends on a function
$\mathsf{h}:\mathsf{\Sigma}\to\mathbb{C}$ used to describe its jump. For the
moment we assume

$\mathsf{h}\in
C^{\infty}(\mathsf{\Sigma}),\quad\mathsf{h}(z)\in\mathbb{R}\text{ for
}z\in\mathbb{R},\quad\text{and}\quad\liminf_{\begin{subarray}{c}z\to\infty\\\
z\in\mathsf{\Sigma}\end{subarray}}\frac{\operatorname{Re}\mathsf{h}(z)}{|z|}>0.$

These conditions are present only to ensure the RHP below is well posed and
are far from optimal, but enough for our purposes. Later on we will impose
more conditions on this function $\mathsf{h}$, these conditions will be
tailored to our later needs regarding the asymptotic analysis of OPs.

The associated RHP asks for finding a $2\times 2$ matrix-valued function
$\mathbf{\Phi}$ with the following properties.

  1. $\mathbf{\Phi}$-1.

The matrix
$\mathbf{\Phi}=\mathbf{\Phi}(\cdot\mid\mathsf{h}):\mathbb{C}\setminus\mathsf{\Sigma}\to\mathbb{C}^{2\times
2}$ is analytic.

  2. $\mathbf{\Phi}$-2.

Along the interior of the arcs of $\mathsf{\Sigma}$ the function
$\mathbf{\Phi}$ admits continuous boundary values $\mathbf{\Phi}_{\pm}$
related by
$\mathbf{\Phi}_{+}(\zeta)=\mathbf{\Phi}_{-}(\zeta)\mathbf{J}_{\mathbf{\Phi}}(\zeta)$,
$\zeta\in\mathsf{\Sigma}$, with

$\mathbf{J}_{\mathbf{\Phi}}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\mathbf{I}+\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{h}(\zeta)}}\mathbf{E}_{12},&\zeta\in\mathsf{\Sigma}_{0},\\\
\mathbf{I}+(1+\operatorname{\rm
e}^{-\mathsf{h}(\zeta)})\mathbf{E}_{21},&\zeta\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3},\\\
\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{h}(\zeta)}}\mathbf{E}_{12}-(1+\operatorname{\rm
e}^{-\mathsf{h}(\zeta)})\mathbf{E}_{21},&\zeta\in\mathsf{\Sigma}_{2}.\end{dcases}$
(4.2)

  3. $\mathbf{\Phi}$-3.

As $\zeta\to\infty$,

$\mathbf{\Phi}(\zeta)=\left(\mathbf{I}+\frac{1}{\zeta}\mathbf{\Phi}^{(1)}+\mathcal{O}(1/\zeta^{2})\right)\zeta^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\operatorname{\rm
e}^{-\frac{2}{3}\zeta^{3/2}\bm{\sigma}_{3}},$ (4.3)

where

$\mathbf{U}_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{pmatrix}$ (4.4)

and $\mathbf{\Phi}^{(1)}=\mathbf{\Phi}^{(1)}(\mathsf{h})$ is a matrix that
depends on the choice of function $\mathsf{h}$ but it is independent of
$\zeta$.

  4. $\mathbf{\Phi}$-4.

The matrix $\mathbf{\Phi}$ remains bounded as $\zeta\to 0$.

Given $\mathsf{h}$, it is not at all obvious that the RHP above has a solution
and how to describe it. We study this model problem when
$\mathsf{h}=\mathsf{h}_{\tau}$ depends on an additional large parameter
$\tau$, in a way that appears naturally in the asymptotic analysis of the
orthogonal polynomials mentioned earlier. For large values of $\tau$, we then
prove that the solution $\mathbf{\Phi}$ exists and is asymptotically close to
a model RHP that appeared recently [24] and that we discuss in a moment.

###  4.2. The model RHP with admissible data

For us, we need to consider the model problem
$\mathbf{\Phi}=\mathbf{\Phi}(\cdot\mid\mathsf{h})$ with functions
$\mathsf{h}=\mathsf{h}_{\tau}$ satisfying certain properties which are
formally introduced in the next definition.

######  Definition 4.1.

We call a function $\mathsf{h}_{\tau}:\Sigma\to\mathbb{C}$ admissible if it is
of the form

$\mathsf{h}_{\tau}(\zeta)=\mathsf{h}_{\tau}(\zeta\mid\mathsf{s})=\mathsf{s}+\tau\mathsf{H}\left(\frac{\mathsf{\zeta}}{\tau}\right),\quad\zeta\in\mathbb{C},\quad\tau>0,\;\mathsf{s}\in\mathbb{R},$

where $\mathsf{H}$ is defined on a neighborhood $\mathcal{S}$ of $\Sigma$ and
satisfies the following properties.

  1. (i)

The function $\mathsf{H}$ is independent of $\tau$ and $\mathsf{s}$, of class
$C^{\infty}$ on $\mathcal{S}$ and real-valued along $\mathbb{R}$.

  2. (ii)

$\mathsf{H}$ is analytic on a disk $D_{\delta}(0)\subset\mathcal{S}$ centered
at the origin, and its unique zero on $D_{\delta}(0)$ is at $\zeta=0$, with

$\mathsf{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\mathsf{H}^{\prime}(0)>0.$

  3. (iii)

There exist constants $\eta,\widehat{\eta}>0$ for which

$\operatorname{Re}\mathsf{H}(w)>\eta|w|\quad\text{for
}w\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{2}\cup\mathsf{\Sigma}_{3},$

and

$-\widehat{\eta}w^{3/2-\epsilon}<\mathsf{H}(w)<-\eta w\quad\text{for
}w\in\mathsf{\Sigma}_{0},$

for some $\epsilon\in(0,1/2]$.

Conditions (i)–(ii), and also the bounds in (iii) involving $\eta$, are
natural in our setup. The bound
$\mathsf{H}(w)>-\widehat{\eta}w^{3/2-\epsilon}$ is present for technical
reasons, and it plays a role only for the proof of Lemma 7.4, allowing us to
write certain estimates in a cleaner matter. It could be removed, at the cost
of slightly more complicated error terms in the mentioned Lemma. For our
purposes, namely to use
$\mathbf{\Phi}=\mathbf{\Phi}(\cdot\mid\mathsf{h}_{\tau})$ as a local
parametrix with an appropriate $\mathsf{h}_{\tau}$, this condition is
satisfied anyway (this will be accomplished in Proposition 8.3), so we include
it in our definition here as well, as it simplifies our analysis.

In the course of the analysis for the RHP for the orthogonal polynomials
discussed in Section 9, the function $\mathsf{H}$ will be a transformation of
the function $Q$ appearing in the deformed weight (2.17), and the parameter
$\mathsf{t}$ that we defined here will play the same role as the one in the
definition (2.7).

Given an admissible $\mathsf{h}_{\tau}$, we denote

$\mathbf{\Phi}_{\tau}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{\Phi}(\zeta\mid\mathsf{h}=\mathsf{h}_{\tau}(\cdot\mid\mathsf{s})).$
(4.5)

We are interested in the asymptotic analysis for $\mathbf{\Phi}_{\tau}$ as
$\tau\to+\infty$ and $\mathsf{s}\geq-\mathsf{s}_{0}$, for any
$\mathsf{s}_{0}>0$, and $\mathsf{t}>0$ kept fixed within a compact of the
positive axis.

We now explain in an ad hoc manner the appearance of a RHP for the integro-
differential equation, which also relates to the KPZ equation. Definition
4.1–(ii) gives that $\mathsf{H}$ has an expansion of the form

$\mathsf{H}(\zeta)=-\mathsf{t}\zeta(1+\mathcal{O}(\zeta)),\quad|\zeta|\leq\delta,$

This means that any admissible function $\mathsf{h}_{\tau}$ satisfies

$\mathsf{h}_{\tau}(\zeta)=\mathsf{s}-\mathsf{t}\zeta\left(1+\mathcal{O}(\zeta\tau^{-1})\right),\quad|\zeta|\leq\delta\tau.$

In particular, the convergence

$\mathsf{h}_{\tau}(\zeta)\to\mathsf{h}_{0}(\zeta)=\mathsf{h}_{0}(\zeta\mid\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathsf{s}-\mathsf{t}\zeta,$
(4.6)

holds true uniformly in compacts as $\tau\to\infty$. This indicates that the
solution $\mathbf{\Phi}_{\tau}$ should converge to the solution

${\mathbf{\Phi}}_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{\Phi}(\cdot\mid\mathsf{h}=\mathsf{h}_{0})$
(4.7)

of the model problem obtained from $\mathsf{h}_{0}$. The
RHP-${\mathbf{\Phi}}_{0}$ relates to the integro-differential PII and is a
rescaled version of an RHP that appears in the description of the narrow wedge
solution to the KPZ equation, as we discuss in the next section in detail.

##  5\. The RHP for the integro-differential RHP

For the choice

$\mathsf{h}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta)=\mathsf{h}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta\mid
s,T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-T^{1/3}(s+\zeta)$
(5.1)

the corresponding solution of the RHP–$\mathbf{\Phi}$

${\mathbf{\Phi}}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta)={\mathbf{\Phi}}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta\mid
s,T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{\Phi}(\zeta\mid\mathsf{h}=\mathsf{h}^{\mathrm{\scriptscriptstyle(KPZ)}}(\cdot\mid
s,T))$

appeared for the first time in the work of Cafasso and Clayes [24] (this is
the RHP-$\Psi$ in Section 2 therein) in connection with the narrow wedge
solution to the KPZ equation as we explain in a moment, in Section 5.1. To
avoid confusion with the related quantities that we are about to introduce, we
term it the KPZ RHP. In virtue of the identity

$\mathsf{h}_{0}(\zeta\mid\mathsf{s},\mathsf{t})=\mathsf{h}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta\mid
s=-\mathsf{s}/\mathsf{t},T=\mathsf{t}^{3})$

which follows from (4.6) and (5.1), we also have the correspondence

${\mathbf{\Phi}}_{0}(\zeta\mid\mathsf{s},\mathsf{t})={\mathbf{\Phi}}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta\mid
s=-\mathsf{s}/\mathsf{t},T=\mathsf{t}^{3}),$ (5.2)

and we refer to ${\mathbf{\Phi}}_{0}$ as the id-PII RHP. For the record, we
state the existence of ${\mathbf{\Phi}}_{0}$ formally as a result.

######  Proposition 5.1.

For any $\mathsf{s}\in\mathbb{R}$ and any $\mathsf{t}>0$, the solution
${\mathbf{\Phi}}_{0}$ exists and is unique. Furthermore, for any fixed
$\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$, the solution
${\mathbf{\Phi}}_{0,+}(\zeta)$ remains bounded for $\zeta$ in compacts of
$\mathbb{R}$ and $\mathsf{s}\geq-\mathsf{s}_{0}$,
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$.

###### Proof.

It is a consequence of [24, Section 2] that the solution
${\mathbf{\Phi}}^{\mathrm{\scriptscriptstyle(KPZ)}}(\cdot\mid s,T)$ exists and
is unique, for any $s\in\mathbb{R}$ and $T>0$, and from the correspondence
(5.2) the existence and uniqueness of ${\mathbf{\Phi}}_{0}$ is thus granted.

For the boundedness, we start from the representation

${\mathbf{\Phi}}_{0}(\zeta)=\mathbf{I}+\frac{1}{2\pi\mathrm{i}}\int_{\Gamma}\frac{{\mathbf{\Phi}}_{0,-}(s)(\mathbf{J}_{{\mathbf{\Phi}}_{0}}(s)-\mathbf{I})}{s-\zeta}\mathrm{d}s,\quad\zeta\in\mathbb{C}\setminus\Gamma,$

which follows from the $L^{p}$ theory of RHPs (see [34]). The jump matrix
admits an analytic continuation to any neighborhood of the real axis, and this
analytic continuation remains bounded in compacts, also uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$ (see for instance (5.9) for the exact expression). With
these observations in mind, the claimed boundedness follows from standard
arguments. We skip additional details, but refer to the proof of Theorem 7.1,
in particular (7.16) et seq., for similar arguments in a more involved
context. ∎

In this section we collect several results on ${\mathbf{\Phi}}_{0}$ that were
obtained in [24, 25] and which will be needed later.

But before proceeding, a word of caution. As we said, the
RHP–${\mathbf{\Phi}}^{\mathrm{\scriptscriptstyle(KPZ)}}$ appeared first in
[24], but was also studied in the subsequent work [25]. The meanings for the
variables $s,x$ and $t$ in these two works are not consistent, but we need
results from both of them. Comparing to the work [24] by Cafasso and Claeys,
the correspondence is

$s_{\rm CC}=-\frac{\mathsf{s}}{\mathsf{t}}\quad\text{and}\quad T_{\rm
CC}=\mathsf{t}^{3}.$ (5.3)

This correspondence is consistent with (5.2). On the other hand, when
comparing to the subsequent work [25] by Cafasso, Claeys and Ruzza, the
correspondence between notations is

$t_{\rm CCR}=\frac{1}{\mathsf{t}^{3/2}},\quad x_{\rm
CCR}=-\frac{\mathsf{s}}{\mathsf{t}^{3/2}},\qquad\text{that is}\qquad x_{\rm
CCR}=-\mathsf{S},\quad t_{\rm CCR}=\mathsf{T},$ (5.4)

where $\mathsf{T},\mathsf{S}$ are as in (2.18).

In our asymptotic analysis, the most convenient choice of variables to work
with is the choice $(\mathsf{s},\mathsf{t})$ and the correspondence
$(\mathsf{S},\mathsf{T})$ from (2.18) that we have already been using, and
which leads to the RHP ${\mathbf{\Phi}}_{0}$ as we introduced. Nevertheless,
we will need to collect results from both mentioned works, and when the need
arises we refer to the correspondences of variables (5.3)–(5.4).

On the other hand, when making correspondence with integrable systems, in
particular the integro-differential Painlevé II equation, it is more
convenient to work with the variables $\mathsf{S}$ and $\mathsf{T}$ as in
(2.18).

###  5.1. Properties of the id-PII parametrix

In this section we describe many of the findings from [24, 25], in a way
suitably adapted to our notation and needs. In particular the connection of
${\mathbf{\Phi}}_{0}$ introduced in (4.7) with the integro-differential
Painlevé II equation is described in this section.

For

${\mathbf{\Delta}}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta)={\mathbf{\Delta}}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta\mid
s,T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\mathbf{I},&\zeta>0,\\\
\mathbf{I}+(1+\operatorname{\rm
e}^{T^{1/3}(\zeta+s)})\mathbf{E}_{21},&\zeta<0.\end{dcases}$ (5.5)

the identity

$\partial_{s}\log\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s,T)=\frac{T^{1/3}}{2\pi\mathrm{i}}\int_{-\infty}^{\infty}\frac{\operatorname{\rm
e}^{T^{1/3}(x+s)}}{(1+\operatorname{\rm
e}^{T^{1/3}(x+s)})^{2}}\left[({\mathbf{\Delta}}^{\mathrm{\scriptscriptstyle(KPZ)}}(x)^{-1}{\mathbf{\Phi}}^{\mathrm{\scriptscriptstyle(KPZ)}}_{+}(x)^{-1}({\mathbf{\Phi}}^{\mathrm{\scriptscriptstyle(KPZ)}}_{+}{\mathbf{\Delta}}^{\mathrm{\scriptscriptstyle(KPZ)}})^{\prime}(x)\right]_{21}\mathrm{d}x,$
(5.6)

was shown in [24, Theorem 2.1] and will also be useful for us. With (5.2) we
now rewrite this identity in terms of ${\mathbf{\Phi}}_{0}$. With the
principal branch of the argument, set

${\mathbf{\Delta}}_{0}(\zeta)={\mathbf{\Delta}}_{0}(\zeta\mid\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{cases}\mathbf{I},&|\arg\zeta|<\frac{2\pi}{3},\\\
\mathbf{I}+(1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta})\mathbf{E}_{21},&|\arg\zeta|>\frac{2\pi}{3}.\end{cases}$
(5.7)

This function relates to
${\mathbf{\Delta}}^{\mathrm{\scriptscriptstyle(KPZ)}}$ in (5.5) via

${\mathbf{\Delta}}_{0,+}(\zeta)={\mathbf{\Delta}}^{\mathrm{\scriptscriptstyle(KPZ)}}(\zeta\mid
s=-\mathsf{s}/\mathsf{t},T=\mathsf{t}^{3}),\quad\zeta\in\mathbb{R},$

and (5.6) rewrites as

$\partial_{s}\log\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s=-\mathsf{s}/\mathsf{t},T=\mathsf{t}^{3})=\frac{\mathsf{t}}{2\pi\mathrm{i}}\int_{-\infty}^{\infty}\frac{\operatorname{\rm
e}^{\mathsf{t}x-\mathsf{s}}}{(1+\operatorname{\rm
e}^{\mathsf{t}x-\mathsf{s}})^{2}}\left[({\mathbf{\Delta}}_{0}(x)^{-1}{\mathbf{\Phi}}_{0,+}(x)^{-1}({\mathbf{\Phi}}_{0,+}{\mathbf{\Delta}}_{0})^{\prime}(x)\right]_{21}\mathrm{d}x.$
(5.8)

For further reference, it is now convenient to state the RHP for
${\mathbf{\Phi}}_{0}$ explicitly.

  1. ${\mathbf{\Phi}}_{0}$-1.

The matrix
${\mathbf{\Phi}}_{0}:\mathbb{C}\setminus\mathsf{\Sigma}\to\mathbb{C}^{2\times
2}$ is analytic.

  2. ${\mathbf{\Phi}}_{0}$-2.

Along the interior of the arcs of $\mathsf{\Sigma}$ the function
${\mathbf{\Phi}}_{0}$ admits continuous boundary values
${\mathbf{\Phi}}_{0,\pm}$ related by
${\mathbf{\Phi}}_{0,+}(\zeta)={\mathbf{\Phi}}_{0,-}(\zeta)\mathbf{J}_{{\mathbf{\Phi}}_{0}}(\zeta)$,
$\zeta\in\mathsf{\Sigma}$, with

$\mathbf{J}_{{\mathbf{\Phi}}_{0}}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\mathbf{I}+\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta}}\mathbf{E}_{12},&\zeta\in\mathsf{\Sigma}_{0},\\\
\mathbf{I}+(1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta})\mathbf{E}_{21},&\zeta\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3},\\\
\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta}}\mathbf{E}_{12}-(1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta})\mathbf{E}_{21},&\zeta\in\mathsf{\Sigma}_{2}.\end{dcases}$
(5.9)

  3. ${\mathbf{\Phi}}_{0}$-3.

As $\zeta\to\infty$,

${\mathbf{\Phi}}_{0}(\zeta)=\left(\mathbf{I}+\mathcal{O}(1/\zeta)\right)\zeta^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\operatorname{\rm
e}^{-\frac{2}{3}\zeta^{3/2}\sigma_{3}}.$ (5.10)

where we recall that $\mathbf{U}_{0}$ is given in (4.4).

  4. ${\mathbf{\Phi}}_{0}$-4.

The matrix ${\mathbf{\Phi}}_{0}$ remains bounded as $\zeta\to 0$.

To compare with [25] we perform a transformation of this RHP. All the
calculations that follow already take into account the correspondence (5.4)
between the notation in the mentioned work and our notation.

With ${\mathbf{\Delta}}_{0}$ as in (5.7) and introducing

$\xi=\xi(\zeta)=-\mathsf{s}+\mathsf{t}\zeta,\quad\text{with
inverse}\quad\zeta=\zeta(\xi)=\frac{\xi+\mathsf{s}}{\mathsf{t}},$

we transform

${\mathbf{\Psi}}_{0}(\xi)=\left(\mathbf{I}+\frac{\mathrm{i}\mathsf{s}^{2}}{4\mathsf{t}^{3/2}}\mathbf{E}_{12}\right)\mathsf{t}^{\bm{\sigma}_{3}/4}{\mathbf{\Phi}}_{0}(\zeta(\xi))\times\begin{cases}{\mathbf{\Delta}}_{0}(\zeta(\xi)),&\operatorname{Im}\xi>0,\;\arg(\zeta(\xi))\neq
2\pi/3,\\\
{\mathbf{\Delta}}_{0}(\zeta(\xi))^{-1},&\operatorname{Im}\xi<0,\;\arg(\zeta(\xi))\neq-2\pi/3.\end{cases}$
(5.11)

Then ${\mathbf{\Psi}}_{0}$ satisfies the following RHP.

  1. ${\mathbf{\Psi}}_{0}$-1.

The matrix
${\mathbf{\Psi}}_{0}:\mathbb{C}\setminus\mathbb{R}\to\mathbb{C}^{2\times 2}$
is analytic.

  2. ${\mathbf{\Psi}}_{0}$-2.

Along $\mathbb{R}$ the function ${\mathbf{\Psi}}_{0}$ admits continuous
boundary values ${\mathbf{\Psi}}_{0,\pm}$ related by

${\mathbf{\Psi}}_{0,+}(\xi)={\mathbf{\Psi}}_{0,-}(\xi)\left(\mathbf{I}+\frac{1}{1+\operatorname{\rm
e}^{\xi}}\mathbf{E}_{12}\right),\quad\xi\in\mathbb{R}.$

  3. ${\mathbf{\Psi}}_{0}$-3.

For any $\delta\in(0,2\pi/3)$, as $\xi\to\infty$ the matrix
${\mathbf{\Psi}}_{0}$ has the following asymptotic behavior,

${\mathbf{\Psi}}_{0}(\xi)=\left(\mathbf{I}+\mathcal{O}(1/\xi)\right)\xi^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\operatorname{\rm
e}^{-\mathsf{t}^{-3/2}\left(\frac{2}{3}\xi^{3/2}+\mathsf{s}\xi^{1/2}\right)\sigma_{3}}\times\begin{cases}\mathbf{I},&|\arg\xi|\leq\pi-\delta,\\\
\mathbf{I}\pm\mathbf{E}_{21},&\pi-\delta<\pm\arg\xi<\pi.\end{cases}$ (5.12)

This RHP is the same RHP considered in [25, page 1120]111The keen reader will
notice that there is a sign difference between the last term in the right-hand
side of (4.3) and the corresponding term in [25, page 1120], but the latter is
a typo. with the choice $\sigma(r)=(1+\operatorname{\rm e}^{-r})^{-1}$ therein
and the correspondence of variables (5.4).

As a consequence, and with the change of variables
$(\mathsf{s},\mathsf{t})\mapsto(\mathsf{S},\mathsf{T})$ from (2.18), we obtain
that for some functions
$\mathsf{Q}=\mathsf{Q}(\mathsf{S},\mathsf{T}),\mathsf{R}=\mathsf{R}(\mathsf{S},\mathsf{T}),\mathsf{P}=\mathsf{P}(\mathsf{S},\mathsf{T})$
and

$\mathsf{q}=\mathsf{q}(\mathsf{s},\mathsf{t})=\mathsf{Q}(\mathsf{S},\mathsf{T}),\quad\mathsf{r}=\mathsf{r}(\mathsf{s},\mathsf{t})=\mathsf{R}(\mathsf{S},\mathsf{T}),\quad\mathsf{p}=\mathsf{p}(\mathsf{s},\mathsf{t})=\mathsf{P}(\mathsf{S},\mathsf{T}),$
(5.13)

the asymptotic behavior (5.12) improves to

${\mathbf{\Psi}}_{0}(\xi)=\left(\mathbf{I}+\frac{1}{\xi}\begin{pmatrix}\mathsf{q}&-\mathrm{i}\mathsf{r}\\\
\mathrm{i}\mathsf{p}&-\mathsf{q}\end{pmatrix}+\mathcal{O}(\xi^{-2})\right)\xi^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\\\
\times\operatorname{\rm
e}^{-\mathsf{t}^{-3/2}\left(\frac{2}{3}\xi^{3/2}+\mathsf{s}\xi^{1/2}\right)\sigma_{3}}\times\begin{cases}\mathbf{I},&|\arg\xi|\leq\pi-\delta,\\\
\mathbf{I}\pm\mathbf{E}_{21},&\pi-\delta<\pm\arg\xi<\pi,\end{cases}\quad\xi\to\infty.$
(5.14)

Stressing that the correspondence (5.4) is in place, the functions
$\mathsf{P}$ and $\mathsf{Q}$ satisfy the relation [25, Equation (3.14)]

$\partial_{\mathsf{S}}\mathsf{P}(\mathsf{S},\mathsf{T})=2\mathsf{Q}(\mathsf{S},\mathsf{T})+\mathsf{P}(\mathsf{S},\mathsf{T})^{2}.$

Furthermore, from [25, Equations (3.12),(3.16), Theorem 1.3 and Corollary 1.4]
we see that ${\mathbf{\Psi}}_{0}$ takes the form

${\mathbf{\Psi}}_{0}(\xi\mid\mathsf{s},\mathsf{t})=\sqrt{2\pi}\operatorname{\rm
e}^{-\frac{\pi\mathrm{i}}{4}\bm{\sigma}_{3}}\left(\mathbf{I}-\mathsf{p}(\mathsf{s},\mathsf{t})\mathbf{E}_{12}\right)\begin{pmatrix}-\partial_{\mathsf{S}}\Phi(\xi\mid\mathsf{S},\mathsf{T})&\ast\\\
-\Phi(\xi\mid\mathsf{S},\mathsf{T})&\ast\end{pmatrix}\operatorname{\rm
e}^{\frac{\pi\mathrm{i}}{4}\bm{\sigma}_{3}},$ (5.15)

where $\Phi=\Phi(\xi\mid\mathsf{S},\mathsf{T})$ solves the NLS equation with
potential $2\partial_{\mathsf{S}}\mathsf{P}$,

$\partial_{\mathsf{S}}^{2}\Phi(\xi\mid\mathsf{S},\mathsf{T})=(\xi+2\partial_{\mathsf{S}}\mathsf{P}(\mathsf{S},\mathsf{T}))\Phi(\xi\mid\mathsf{S},\mathsf{T}).$

In addition, $\mathsf{P}$ and $\Phi$ are related through the identity (2.22)
which, in turn, implies that $\Phi$ is the solution to the integro-
differential Painlevé II equation in (2.11).

It is convenient to write some quantities of ${\mathbf{\Phi}}_{0}$ directly in
terms of the just introduced functions. The first identity we need for later
is

$\left[\left({\mathbf{\Phi}}_{0}(\zeta(v)\mid\mathsf{s},\mathsf{t}){\mathbf{\Delta}}_{0}(\zeta(v)\mid\mathsf{s},\mathsf{t})\right)^{-1}{\mathbf{\Phi}}_{0}(\zeta(u)\mid\mathsf{s},\mathsf{t}){\mathbf{\Delta}}_{0}(\zeta(u)\mid\mathsf{s},\mathsf{t})\right]_{21,+}\\\
=-2\pi\mathrm{i}\left(\Phi(u\mid\mathsf{S},\mathsf{T})(\partial_{\mathsf{S}}\Phi)(v\mid\mathsf{S},\mathsf{T})-\Phi(v\mid\mathsf{S},\mathsf{T})(\partial_{\mathsf{S}}\Phi)(u\mid\mathsf{S},\mathsf{T})\right)$
(5.16)

which follows from (5.11) and (5.15) after a straightforward calculation,
accounting also that $\det{\mathbf{\Phi}}_{0}=\det{\mathbf{\Psi}}_{0}\equiv
1$.

The second relation we need is an improvement of the asymptotics of
${\mathbf{\Phi}}_{0}$ in (5.10). With the coefficients

$\mathsf{c}_{1}=\mathsf{c}_{1}(\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{\mathsf{s}^{2}}{4\mathsf{t}^{3/2}},\quad\mathsf{c}_{2}=\mathsf{c}_{2}(\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\mathsf{s}^{4}}{32\mathsf{t}^{3}},\quad\mathsf{c}_{3}=\mathsf{c}_{3}(\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{\mathsf{s}^{3}(\mathsf{s}^{3}-16\mathsf{t}^{3})}{384\mathsf{t}^{9/2}},$

and the functions $\mathsf{q},\mathsf{r},\mathsf{p}$ in (5.13), introduce

${\mathbf{\Phi}}_{0}^{(1)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{\mathsf{t}}\begin{pmatrix}-\dfrac{\mathsf{s}}{4}+\mathsf{q}+\mathsf{c}_{2}-\mathsf{c}_{1}\mathsf{p}-\mathsf{c}_{1}^{2}&\mathrm{i}\mathsf{t}^{-1/2}\left(-\mathsf{r}-2\mathsf{q}\mathsf{c}_{1}+\dfrac{\mathsf{s}}{2}\mathsf{c}_{1}+\mathsf{p}\mathsf{c}_{1}^{2}+\mathsf{c}_{1}\mathsf{c}_{2}-\mathsf{c}_{3}\right)\\\
\mathrm{i}\mathsf{t}^{1/2}(\mathsf{p}+\mathsf{c}_{1})&\dfrac{\mathsf{s}}{4}-\mathsf{q}+\mathsf{p}\mathsf{c}_{1}+\mathsf{c}_{2}\end{pmatrix}.$
(5.17)

After some cumbersome but straightforward calculations, the asymptotics (5.14)
improves (5.10) to

${\mathbf{\Phi}}_{0}(\zeta)=\left(\mathbf{I}+\frac{{\mathbf{\Phi}}_{0}^{(1)}}{\zeta}+\mathcal{O}(\zeta^{-2})\right)\zeta^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\operatorname{\rm
e}^{-\frac{2}{3}\zeta^{3/2}\bm{\sigma}_{3}},\quad\zeta\to\infty.$

##  6\. Bounds on the id-PII RHP

We need to obtain certain asymptotic bounds on ${\mathbf{\Phi}}_{0}$ in
different regimes. These bounds will be used later to show that the model
problem $\mathbf{\Phi}_{\tau}$ converges, as $\tau\to+\infty$, to
${\mathbf{\Phi}}_{0}$ as already indicated in (4.5) et seq. We split these
necessary estimates in the next subsections, depending on the regime we are.

In what follows, for a matrix-valued function $\mathbf{M}=(\mathbf{M}_{jk})$
and a contour $\Sigma\subset\mathbb{C}$, we also use the pointwise matrix norm

$|\mathbf{M}(\zeta)|\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max_{j,k}|\mathbf{M}_{j,k}(\zeta)|,$
(6.1)

and the matrix $L^{p}$ norm (possibly also with $p=\infty$)

$\|\mathbf{M}\|_{L^{p}(\Sigma)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max_{j,k}\|\mathbf{M}_{j,k}\|_{L^{p}(\Sigma)},$
(6.2)

where the measure is always understood to be the arc-length measure. In
particular, for any two given matrices $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$
the inequality

$\|\mathbf{M}_{1}\mathbf{M}_{2}\|_{L^{\infty}(\Sigma)}\leq
2\|\mathbf{M}_{1}\|_{L^{\infty}(\Sigma)}\|\mathbf{M}_{2}\|_{L^{\infty}(\Sigma)}$

is satisfied. Similar straightforward inequalities involving $L^{1},L^{2}$ and
$L^{\infty}$ and the pointwise norm (6.1) also hold, and will be used without
further mention. Sometimes we also write

$\|\mathbf{M}\|_{L^{p}\cap
L^{q}(\Sigma)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max\left\\{\|\mathbf{M}\|_{L^{p}(\Sigma)},\|\mathbf{M}\|_{L^{q}(\Sigma)}\right\\},$
(6.3)

to identify that possible convergences are taking place in various norms
simultaneously. In a similar manner we define norms $\|\cdot\|_{L^{p_{1}}\cap
L^{p_{2}}\cap L^{p_{3}}}$ involving three (or possibly more) function spaces.

###  6.1. The singular regime

The first asymptotic regime we consider is

$\mathsf{s}\geq\mathsf{s}_{0}\quad\text{and}\quad\mathsf{t}_{0}\leq\mathsf{t}\leq\frac{1}{\mathsf{t}_{0}},$

where $\mathsf{t}_{0}\in(0,1)$ is any given value, and
$\mathsf{s}_{0}=\mathsf{s}_{0}(\mathsf{t}_{0})>0$ will be made sufficiently
large depending on $\mathsf{t}_{0}>0$, but independent of $\mathsf{t}$ within
the range above. With (5.4) in mind, this is a particular case of the singular
regime in [25].

For this asymptotic regime, we need the following result.

######  Proposition 6.1.

For any $\mathsf{t}_{0}\in(0,1)$ there exists
$\mathsf{s}_{0}=\mathsf{s}_{0}(\mathsf{t}_{0})>0$, $M=M(\mathsf{t}_{0})>0$ and
$\eta=\eta(\mathsf{t}_{0})>0$ such that the inequalities

$\displaystyle\left|{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{12}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}\right|\leq
M\operatorname{\rm e}^{-\eta\operatorname{Re}(\zeta^{3/2})},$
$\displaystyle\zeta\in\mathsf{\Sigma}_{0},$
$\displaystyle\left|{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{21}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}\right|\leq
M\operatorname{\rm e}^{-\eta\operatorname{Re}(\zeta^{3/2})},$
$\displaystyle\zeta\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3},\quad\text{and}$
$\displaystyle\left|{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{22}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}\right|\leq
M|\zeta|^{1/2},$ $\displaystyle\zeta\in\mathsf{\Sigma}_{2},$

hold true for any $\mathsf{s}\geq\mathsf{s}_{0}$ and any
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

The proof of Proposition 6.2 is a recollection of the analysis in [25], so
before going into the details we need to review some further notions from
their work.

Introduce

${\mathbf{\Phi}}_{\rm
Ai}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\sqrt{2\pi}\times\begin{cases}\begin{pmatrix}\operatorname{Ai}^{\prime}(\zeta)&-\operatorname{\rm
e}^{2\pi\mathrm{i}/3}\operatorname{Ai}^{\prime}(\operatorname{\rm
e}^{-2\pi\mathrm{i}/3}\zeta)\\\
\mathrm{i}\operatorname{Ai}(\zeta)&-\mathrm{i}\operatorname{\rm
e}^{-2\pi\mathrm{i}/3}\operatorname{Ai}(\operatorname{\rm
e}^{-2\pi\mathrm{i}/3}\zeta)\end{pmatrix},&\operatorname{Im}\zeta>0,\\\
\begin{pmatrix}\operatorname{Ai}^{\prime}(\zeta)&\operatorname{\rm
e}^{-2\pi\mathrm{i}/3}\operatorname{Ai}^{\prime}(\operatorname{\rm
e}^{2\pi\mathrm{i}/3}\zeta)\\\
\mathrm{i}\operatorname{Ai}(\zeta)&\mathrm{i}\operatorname{\rm
e}^{2\pi\mathrm{i}/3}\operatorname{Ai}(\operatorname{\rm
e}^{2\pi\mathrm{i}/3}\zeta)\end{pmatrix},&\operatorname{Im}\zeta<0.\end{cases}$
(6.4)

This is the matrix appearing in [25, Equation (2.5)]. With the correspondence
of variables (5.4) in mind, when we combine our identity (5.11) with [25,
Equation (2.8)], we obtain the equality

${\mathbf{\Phi}}_{0}(\zeta)=\mathbf{Y}(\zeta){\mathbf{\Phi}}_{\rm
Ai}(\zeta)\times\begin{cases}\mathbf{I},&-2\pi/3<\arg\zeta<2\pi/3,\\\
\mathbf{I}\mp(1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta})\mathbf{E}_{21},&2\pi/3<\pm\arg\zeta<\pi.\end{cases}$
(6.5)

The exact form of the matrix $\mathbf{Y}$ is not important for us, but we can
interpret this last equality as a defining identity for $\mathbf{Y}$. What is
important for us is that $\mathbf{Y}$ is analytic off the real axis, with a
jump matrix $\mathbf{J}_{\mathbf{Y}}$ on $\mathbb{R}$ which admits an analytic
continuation to a neighborhood of the axis.

The small norm theory for $\mathbf{Y}$ in our regime of interest was carried
out in [25, Lemma 5.1 and Section 5.2]. As a consequence, we obtain that for
any $\mathsf{t}_{0}>0$ there exist
$M=M(\mathsf{t}_{0})>0,\mathsf{s}_{0}=\mathsf{s}_{0}(\mathsf{t}_{0})>0,\eta=\eta(\mathsf{t}_{0})>0$
such that the inequalities

$\|\mathbf{J}_{\mathbf{Y}}-\mathbf{I}\|_{L^{2}\cap L^{\infty}\cap
L^{1}(\mathbb{R})}\leq M\operatorname{\rm e}^{-\eta
s},\quad\text{and}\quad\|\mathbf{Y}_{\pm}-\mathbf{I}\|_{L^{2}\cap
L^{\infty}\cap L^{1}(\mathbb{R})}\leq M\operatorname{\rm e}^{-\eta s}$

hold for any $\mathsf{s}\geq\mathsf{s}_{0}$ and any
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$. Also as a consequence of the
small norm theory, we obtain the expression

$\mathbf{Y}(\zeta)=\mathbf{I}+\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}}\frac{\mathbf{Y}_{-}(x)(\mathbf{J}_{\mathbf{Y}}(x)-\mathbf{I})}{x-\zeta}\mathrm{d}x,\quad\zeta\in\mathbb{C}\setminus\mathbb{R}.$

We combine this last identity with the fact that $\mathbf{J}_{\mathbf{Y}}$
admits an analytic continuation in a neighborhood of $\mathbb{R}$, and learn
that there exists $M=M(\mathsf{t}_{0})>0$ for which

$|\mathbf{Y}(\zeta)^{\pm 1}|\leq M,$ (6.6)

for every $\zeta\in\mathbb{C}$, $\mathsf{s}\geq\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

###### Proof of Proposition 6.1.

For $\zeta\in\mathsf{\Sigma}_{0}=(0,\infty)$, we use (6.5) and the definition
of ${\mathbf{\Phi}}_{\rm Ai}$ in

${\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{12}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}=2\pi\mathbf{Y}_{+}(\zeta)\begin{pmatrix}-\mathrm{i}\operatorname{Ai}(\zeta)\operatorname{Ai}^{\prime}(\zeta)&\operatorname{Ai}^{\prime}(\zeta)^{2}\\\
\operatorname{Ai}(\zeta)^{2}&\mathrm{i}\operatorname{Ai}(\zeta)\operatorname{Ai}^{\prime}(\zeta)\end{pmatrix}\mathbf{Y}_{+}(\zeta)^{-1}$

Using the bound (6.6), the continuity and the known asymptotics as
$\zeta\to\infty$ of the Airy function and its derivative, the claim along
$\mathsf{\Sigma}_{0}$ follows.

The claim for $\zeta\in\mathsf{\Sigma}_{j}$ with $j=1,2,3$ follows in exactly
the same explicit manner, we skip the details. ∎

###  6.2. The non-asymptotic regime

In the non-asymptotic regime, we fix any $\mathsf{t}_{0}\in(0,1)$ and
$\mathsf{s}_{0}>0$ and seek for bounds of certain entries of
${\mathbf{\Phi}}_{0}$ which are valid uniformly within the range

$|\mathsf{s}|\leq\mathsf{s}_{0}\quad\text{and}\quad\mathsf{t}_{0}\leq\mathsf{t}\leq\frac{1}{\mathsf{t}_{0}}.$

For the next result, we recall the matrix norm introduced in (6.1).

######  Proposition 6.2.

Fix any values $\mathsf{t}_{0}\in(0,1)$ and $\mathsf{s}_{0}>0$. There exist
$M=M(\mathsf{s}_{0},\mathsf{t}_{0})>0$ and
$\eta=\eta(\mathsf{s}_{0},\mathsf{t}_{0})>0$ for which the estimates

$\displaystyle\left|{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{12}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}\right|\leq
M\operatorname{\rm e}^{-\eta\operatorname{Re}(\zeta^{3/2})},$
$\displaystyle\zeta\in\mathsf{\Sigma}_{0},$
$\displaystyle\left|{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{21}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}\right|\leq
M\operatorname{\rm e}^{-\eta\operatorname{Re}(\zeta^{3/2})},$
$\displaystyle\zeta\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3},\quad\text{and}$
$\displaystyle\left|{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{22}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}\right|\leq
M|\zeta|^{1/2},$ $\displaystyle\zeta\in\mathsf{\Sigma}_{2}.$

hold true uniformly for $|\mathsf{s}|\leq\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq\mathsf{t}_{0}^{-1}$.

###### Proof.

The asymptotic behavior as $\zeta\to\infty$ in the RHP–${\mathbf{\Phi}}_{0}$
is valid uniformly up to the boundary values ${\mathbf{\Phi}}_{0,\pm}$ as
well, and also uniformly when the parameters $\mathsf{s}$ and $\mathsf{t}$
vary within compact sets, implying that

$|{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{12}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}|\leq\operatorname{\rm
e}^{-\frac{4}{3}\operatorname{Re}(\zeta^{3/2})}|\zeta^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\mathbf{E}_{12}\mathbf{U}_{0}\zeta^{-\bm{\sigma}_{3}/4}|(1+\mathcal{O}(\zeta^{-1})),\quad\zeta\to\infty.$

Combined with the continuity of the boundary value ${\mathbf{\Phi}}_{0,+}$
with respect to both $\zeta$ and also $\mathsf{s},\mathsf{t}$, the first
estimate follows. The remaining estimates are completely analogous. ∎

##  7\. Asymptotic analysis for the model problem with admissible data

We now carry out the asymptotic analysis as $\tau\to+\infty$ of
$\mathbf{\Phi}_{\tau}$ introduced in (4.5). For that, we fix
$\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$ and work under the assumption
that

$\tau\to+\infty\quad\text{with}\quad\mathsf{s}\geq-\mathsf{s}_{0}\quad\text{and}\quad\mathsf{t}_{0}\leq\mathsf{t}\leq\frac{1}{\mathsf{t}_{0}}.$
(7.1)

During this section, $\mathsf{h}_{\tau}$ always denotes an admissible function
in the sense of Definition 4.1, and $\mathbf{\Phi}_{\tau}$ is the solution to
the associated RHP.

We also talk about uniformity of error terms in the parameter $\mathsf{t}$
ranging on a compact interval $K\subset(0,\infty)$, and by this we mean the
following. The solution $\mathbf{\Phi}_{\tau}$ depends on the parameter
$\mathsf{t}$ via the derivative $\mathsf{H}^{\prime}(0)=-\mathsf{t}$, see
Definition 4.1. We view $\mathsf{H}=\mathsf{H}_{\mathsf{t}}$ as varying with
$\mathsf{t}$ while keeping all the remaining derivatives
$\mathsf{H}^{(k)}(0)$, $k\neq 1$ fixed. By analyticity this determines
$\mathsf{H}$ uniquely at $D_{\delta}(0)$, but not outside this disk. We then
consider $\mathsf{H}_{\mathsf{t}}$ outside $D_{\delta}(0)$ to be any extension
from $D_{\delta}(0)$ that satisfies Definition 4.1 with the additional
requirement that the constants $\eta,\widehat{\eta}$ and $\epsilon$ in (iii)
may depend on $K$ but are independent of $\mathsf{t}\in K$. Of course, for
each $\mathsf{H}_{\mathsf{t}}$ extended this way there corresponds a solution
$\mathbf{\Phi}_{\tau}$ of the associated RHP. By uniformity in $\mathsf{t}\in
K$ we mean that the error may depend on $K$ and the corresponding values
$\eta,\widehat{\eta}$ and $\epsilon$, but is valid for any
$\mathbf{\Phi}_{\tau}$ obtained with an extension $\mathsf{H}_{\mathsf{t}}$
constructed with the explained requirement.

The asymptotic analysis itself makes use of somewhat standard arguments and
objects in the RHP literature. Some consequences of this asymptotic analysis
will be needed later, and we now state them.

The first such consequence is the existence of a solution with asymptotic
formulas relating quantities of interest with the corresponding quantities in
the id-PII RHP.

######  Theorem 7.1.

Fix an admissible function $\mathsf{h}_{\tau}$ in the sense of Definition 4.1.
There exists $\tau_{0}=\tau_{0}(\mathsf{s}_{0},\mathsf{t}_{0})>0$ for which
for any $\tau\geq\tau_{0}$ and any $\mathsf{s},\mathsf{t}$ as in (7.1), the
RHP for $\mathbf{\Phi}(\cdot\mid\mathsf{h}_{\tau})$ admits a unique solution
$\mathbf{\Phi}=\mathbf{\Phi}_{\tau}$ as in (4.5).

Furthermore, for any $\kappa\in(0,1)$, the coefficient
$\mathbf{\Phi}^{(1)}=\mathbf{\Phi}^{(1)}_{\tau}$ in the asymptotic condition
(4.3) satisfies

$\mathbf{\Phi}^{(1)}_{\tau}={\mathbf{\Phi}}_{0}^{(1)}+\mathcal{O}(\tau^{-\kappa}),\quad\tau\to+\infty,$
(7.2)

where $\mathbf{\Phi}^{(1)}_{0}$ is as in (5.17) and the error term is uniform
for $\mathsf{s},\mathsf{t}$ as in (7.1). Also, still for $\kappa\in(0,1)$ the
asymptotic formula

$\mathbf{\Phi}_{\tau,+}(x)=\left(\mathbf{I}+\mathcal{O}\left(\frac{1}{\tau^{\kappa}(1+|x|)}\right)\right){\mathbf{\Phi}}_{0,+}(x),\quad\tau\to+\infty$
(7.3)

holds true uniformly for $x\in\mathsf{\Sigma}$ with
$|x|\leq\tau^{(1-\kappa)/2}$, and uniformly for $\mathsf{s},\mathsf{t}$ as in
(7.1).

The second consequence connects the solution $\mathbf{\Phi}_{\tau}$ directly
with the statistics $\mathsf{Q}$. For its statement, set

$\mathbf{\Delta}_{\tau}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{I}+(1+\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(x)})\chi_{(-\infty,0)}(x)\mathbf{E}_{21},\quad
x\in\mathbb{R}.$

######  Theorem 7.2.

Fix $a,b>0$, $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$. For any
$\kappa\in(0,1)$, the estimate

$\frac{1}{2\pi\mathrm{i}}\int_{\mathsf{s}}^{\infty}\int_{-\tau a}^{\tau
b}\frac{\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}}{\left(1+\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}\right)^{2}}\left[\mathbf{\Delta}_{\tau}(x\mid
u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}\mathrm{d}x\mathrm{d}u=\\\
-\log\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(-\mathsf{s}/\mathsf{t},\mathsf{t}^{3})+\mathcal{O}(\tau^{-\kappa})$

holds as $\tau\to+\infty$, uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$.

For the proof of these results, we compare $\mathbf{\Phi}_{\tau}$ with the
solution ${\mathbf{\Phi}}_{0}$ of the id-PII RHP via the Deift-Zhou nonlinear
steepest descent method. The required asymptotic analysis itself is carried
out Section 7.1, and the proofs of Theorems 7.1 and 7.2 are completed in
Section 7.2.

###  7.1. Asymptotic analysis

For ${\mathbf{\Phi}}_{0}$ as introduced in (5.2) and whose properties were
discussed in Section 5.1, we perform the transformation

$\mathbf{\Psi}_{\tau}(\zeta)=\mathbf{\Phi}_{\tau}(\zeta){\mathbf{\Phi}}_{0}(\zeta)^{-1},\quad\zeta\in\mathbb{C}\setminus\mathsf{\Sigma}.$
(7.4)

Then $\mathbf{\Psi}_{\tau}$ satisfies the following RHP.

  1. $\mathbf{\Psi}_{\tau}$-1.

The matrix
$\mathbf{\Psi}_{\tau}:\mathbb{C}\setminus\mathsf{\Sigma}\to\mathbb{C}^{2\times
2}$ is analytic.

  2. $\mathbf{\Psi}_{\tau}$-2.

Along the interior of the arcs of $\mathsf{\Sigma}$ the function
$\mathbf{\Psi}_{\tau}$ admits continuous boundary values
$\mathbf{\Psi}_{\tau,\pm}$ related by
$\mathbf{\Psi}_{\tau,+}(\zeta)=\mathbf{\Psi}_{\tau,-}(\zeta)\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\zeta)$,
$\zeta\in\mathsf{\Sigma}$. With

$\lambda_{0}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{h}_{0}(\zeta)}},\quad\lambda_{\tau}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{1+\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}},$

where $\mathsf{h}_{0}$ is as in (4.6), the jump matrix
$\mathbf{J}_{\mathbf{\Psi}_{\tau}}$ is

$\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\mathbf{I}+\left(\lambda_{\tau}(\zeta)-\lambda_{0}(\zeta)\right){\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{12}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1},&\zeta\in\mathsf{\Sigma}_{0},\\\
\mathbf{I}+\left(\frac{1}{\lambda_{\tau}(\zeta)}-\frac{1}{\lambda_{0}(\zeta)}\right){\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{21}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1},&\zeta\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3},\\\
\frac{\lambda_{\tau}(\zeta)}{\lambda_{0}(\zeta)}{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{11}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1}+\frac{\lambda_{0}(\zeta)}{\lambda_{\tau}(\zeta)}{\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{22}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1},&\zeta\in\mathsf{\Sigma}_{2}.\end{dcases}$
(7.5)

  3. $\mathbf{\Psi}_{\tau}$-3.

For $\mathbf{\Phi}_{\tau}^{(1)}$ and ${\mathbf{\Phi}}_{0}^{(1)}$ the residues
at $\infty$ of $\mathbf{\Phi}_{\tau}$ and ${\mathbf{\Phi}}_{0}$, respectively,
the matrix $\mathbf{\Psi}_{\tau}$ has the asymptotic behavior

$\mathbf{\Psi}_{\tau}(\zeta)=\mathbf{I}+\frac{1}{\zeta}(\mathbf{\Phi}_{\tau}^{(1)}-{\mathbf{\Phi}}_{0}^{(1)})+\mathcal{O}(1/\zeta^{2})\qquad\text{as}\quad\zeta\to\infty.$
(7.6)

  4. $\mathbf{\Psi}_{\tau}$-4.

The matrix $\mathbf{\Psi}_{\tau}$ remains bounded as $\zeta\to 0$.

The next step is to verify that the jump matrix decays to the identity in the
appropriate norms. The terms in the jump that come from ${\mathbf{\Phi}}_{0}$
are precisely the ones we already estimated in Sections 6.1 and 6.2, so it
remains to estimate the terms involving the $\lambda$-functions. The basic
needed estimate is the following lemma.

######  Lemma 7.3.

Fix $\nu\in(0,1/2)$ and $\mathsf{t}_{0}\in(0,1)$. The estimate

$\operatorname{\rm
e}^{\mathsf{h}_{0}(\zeta)-\mathsf{h}_{\tau}(\zeta)}=1+\mathcal{O}(\zeta^{2}/\tau),\quad\tau\to\infty,$

holds true uniformly for $|\zeta|\leq\tau^{\nu}$ and uniformly for
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$, where the error term is
independent of $\mathsf{s}\in\mathbb{R}$.

###### Proof.

The Definition (4.1) of admissibility of $\mathsf{h}_{\tau}$ ensures that for
$\tau$ sufficiently large, we can expand the term $\mathsf{H}(\zeta/\tau)$ in
power series near the origin and obtain the expansion

$\mathsf{h}_{\tau}(\zeta)=\mathsf{s}-\mathsf{t}\zeta+\mathcal{O}(\zeta^{2}/\tau),\quad\tau\to+\infty,$

valid uniformly for $|\zeta|\leq\tau^{\nu}$, $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$, and with error independent of $\mathsf{s}\in\mathbb{R}$.
Recalling that $\mathsf{h}_{0}(\zeta)=\mathsf{s}-\mathsf{t}\zeta$, the proof
is complete. ∎

We are now able to prove the appropriate convergence of
$\mathbf{J}_{\mathbf{\Psi}_{\tau}}$ to the identity matrix. We split the
analysis into three lemmas, corresponding to different pieces of the contour
$\mathsf{\Sigma}$. In the results that follow we use the matrix norm notations
introduced in (6.1)–(6.3).

######  Lemma 7.4.

Fix $\mathsf{t}_{0}\in(0,1)$, $\mathsf{s}_{0}>0$ and $\nu\in(0,1/2)$. There
exist $\tau_{0}=\tau_{0}(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$,
$M=M(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$ and
$\eta=\eta(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$ for which the inequality

$\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}\cap L^{2}\cap
L^{\infty}(\mathsf{\Sigma}_{0})}\leq M\operatorname{\rm
e}^{-\mathsf{s}}\max\left\\{\tau^{-1+2\nu},\operatorname{\rm
e}^{-\eta\tau^{3\nu/2}}\right\\}$

holds true for any $\tau\geq\tau_{0}$, $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

###### Proof.

Because both $\mathsf{h}_{\tau}$ and $\mathsf{h}_{0}$ are real-valued along
the real line, the inequality

$\left|\lambda_{\tau}(\zeta)-\lambda_{0}(\zeta)\right|=\frac{|\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}-\operatorname{\rm
e}^{-\mathsf{h}_{0}(\zeta)}|}{(1+\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)})(1+\operatorname{\rm
e}^{-\mathsf{h}_{0}(\zeta)})}\leq|\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}-\operatorname{\rm e}^{-\mathsf{h}_{0}(\zeta)}|$

is immediate. For $0\leq\zeta\leq\tau^{\nu}$, we then use Lemma 7.3 and the
explicit expression for $\mathsf{h}_{0}$ in (4.6) and obtain

$\left|\lambda_{\tau}(\zeta)-\lambda_{0}(\zeta)\right|=\mathcal{O}\left(\frac{\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta}}{\tau^{1-2\nu}}\right).$ (7.7)

For $\zeta\geq\tau^{\nu}$, we instead use that both $\mathsf{h}_{\tau}$ and
$\mathsf{h}_{0}$ are real-valued along the positive axis and write

$|\lambda_{\tau}(\zeta)-\lambda_{0}(\zeta)|\leq\left|\lambda_{\tau}(\zeta)-1\right|+\left|\lambda_{\tau}(\zeta)-1\right|=\frac{\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta}}{1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta}}+\frac{\operatorname{\rm
e}^{-\mathsf{s}-\tau\mathsf{H}(\zeta/\tau)}}{1+\operatorname{\rm
e}^{-\mathsf{s}-\tau\mathsf{H}(\zeta/\tau)}}\leq\operatorname{\rm
e}^{-\mathsf{s}}\left(\operatorname{\rm e}^{\mathsf{t}\zeta}+\operatorname{\rm
e}^{-\tau\mathsf{H}(\zeta/\tau)}\right).$

From Definition 4.1–(iii) we bound $\operatorname{\rm
e}^{-\tau\mathsf{H}(\zeta/\tau)}\leq\operatorname{\rm
e}^{\widehat{\eta}\zeta^{3/2-\epsilon}}$ and simplify the last inequality to

$|\lambda_{\tau}(\zeta)-\lambda_{0}(\zeta)|\leq\operatorname{\rm
e}^{-\mathsf{s}}\operatorname{\rm
e}^{\tilde{\eta}\zeta^{\alpha}},\quad\zeta\geq\tau^{\nu},\;\alpha\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max\\{1,3/2-\epsilon\\}<\frac{3}{2},$
(7.8)

for a new value $\tilde{\eta}>0$.

Recall that $\mathbf{J}_{\mathbf{\Psi}_{\tau}}$ was given in (7.5). We use
(7.7) and (7.8) in combination with Propositions 6.1 and 6.2 to get the
existence of a value $\tau_{0}>0$ for which

$|\mathbf{J}_{{\mathbf{\Psi}}_{\tau}}(\zeta)-\mathbf{I}|\leq
M\operatorname{\rm e}^{-\mathsf{s}}\operatorname{\rm
e}^{-\eta\zeta^{3/2}}\left(\chi_{(0,\tau^{\nu})}(\zeta)\frac{\operatorname{\rm
e}^{\mathsf{t}\zeta}}{\tau^{1-2\nu}}+\operatorname{\rm
e}^{\tilde{\eta}\zeta^{\alpha}}\chi_{(\tau^{\nu},+\infty)}(\zeta)\right),\quad\tau\geq\tau_{0},$
(7.9)

where $\eta>0,M>0$ may depend on $\mathsf{s}_{0},\mathsf{t}_{0}$ and
$\nu\in[0,1/2)$, but are independent of $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$. After appropriately
changing the values of $\tilde{\eta},\eta,M$, and having in mind that
$\alpha<3/2$, the result follows from this inequality. ∎

Next, we prove the equivalent result along the pieces of $\mathsf{\Sigma}$
which are not on the real line.

######  Lemma 7.5.

Fix $\mathsf{t}_{0}\in(0,1)$, $\mathsf{s}_{0}>0$ and $\nu\in(0,1/2)$. There
exist $\tau_{0}=\tau_{0}(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$,
$M=M(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$ and
$\eta=\eta(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$, for which the inequality

$\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}\cap L^{2}\cap
L^{\infty}(\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3})}\leq M\operatorname{\rm
e}^{-\mathsf{s}}\max\left\\{\tau^{-1+2\nu},\operatorname{\rm
e}^{-\eta\tau^{3\nu/2}}\right\\}$

holds true for any $\tau\geq\tau_{0}$, $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

###### Proof.

Write

$\frac{1}{\lambda_{\tau}(\zeta)}-\frac{1}{\lambda_{0}(\zeta)}=\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}-\operatorname{\rm
e}^{-\mathsf{h}_{0}(\zeta)}=-\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\zeta}(1-\operatorname{\rm
e}^{\mathsf{h}_{0}(\zeta)-\mathsf{h}_{\tau}(\zeta)}).$

From Lemma 7.3, we estimate for $0\leq|\zeta|\leq\tau^{\nu}$,

$\frac{1}{\lambda_{\tau}(\zeta)}-\frac{1}{\lambda_{0}(\zeta)}=\mathcal{O}\left(\frac{\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}\operatorname{Re}\zeta}}{\tau^{1-2\nu}}\right),\quad\tau\to\infty,$

where the implicit error term is independent of $\mathsf{s}$ and uniform for
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$. On the other hand, from the
explicit form of $\mathsf{h}_{0}$ and Definition 4.1–(iii),

$\left|\frac{1}{\lambda_{\tau}(\zeta)}-\frac{1}{\lambda_{0}(\zeta)}\right|\leq\operatorname{\rm
e}^{-\mathsf{s}}\left(\operatorname{\rm
e}^{\mathsf{t}\operatorname{Re}\zeta}+\operatorname{\rm
e}^{-\eta|\zeta|}\right).$

We combine this inequality with Propositions 6.1 and 6.2 and use them on
(7.5). The conclusion is that there exist $M>0$, $\eta_{1},\eta_{2}>0$ and
$\tau_{0}>0$, depending on $\nu,\mathsf{t}_{0},\mathsf{s}_{0}$, for which the
inequality

$\left|\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\zeta)-\mathbf{I}\right|\leq
M\operatorname{\rm e}^{-\mathsf{s}}\operatorname{\rm
e}^{-\eta_{1}\operatorname{Re}\zeta^{3/2}+\eta_{2}\operatorname{Re}\zeta}\left(\frac{1}{\tau^{1-2\nu}}\chi_{\\{|\zeta|\leq\tau^{\nu}\\}}(\zeta)+\chi_{\\{|\zeta|\geq\tau^{\nu}\\}}(\zeta)\right),\quad\zeta\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3},$
(7.10)

is valid for every $\tau\geq\tau_{0},$ $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$. The definition (4.1) of the
contours $\mathsf{\Sigma}_{1}$ and $\mathsf{\Sigma}_{3}$ assure us that
$\operatorname{Re}\zeta^{3/2}>0$ and $\operatorname{Re}\zeta<0$ on these
contours. After possibly changing the values of the constants
$\eta_{1},\eta_{2}$ and $M$, the result follows. ∎

Finally, we now handle the jump on the negative axis.

######  Lemma 7.6.

Fix $\mathsf{t}_{0}\in(0,1)$, $\mathsf{s}_{0}>0$ and $\nu\in(0,1/2)$. There
exist
$\tau_{0}=\tau_{0}(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0,M=M(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$
and $\eta=\eta(\mathsf{t}_{0},\mathsf{s}_{0},\nu)>0$ for which the inequality

$\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}\cap L^{2}\cap
L^{\infty}(\mathsf{\Sigma}_{2})}\leq M\operatorname{\rm
e}^{-\mathsf{s}}\max\left\\{\tau^{-1+2\nu},\operatorname{\rm
e}^{-\eta\tau^{\nu}}\right\\}$

holds true for any $\tau\geq\tau_{0}$, $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

###### Proof.

The initial step is to rewrite the last line of (7.5) as

$\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\zeta)-\mathbf{I}=\left(\frac{\lambda_{\tau}(\zeta)}{\lambda_{0}(\zeta)}-1\right)\mathbf{I}+\left(\frac{\lambda_{0}(\zeta)}{\lambda_{\tau}(\zeta)}-\frac{\lambda_{\tau}(\zeta)}{\lambda_{0}(\zeta)}\right){\mathbf{\Phi}}_{0,+}(\zeta)\mathbf{E}_{22}{\mathbf{\Phi}}_{0,+}(\zeta)^{-1},\quad\zeta\in\mathsf{\Sigma}_{2}.$
(7.11)

The identities

$\frac{\lambda_{\tau}(\zeta)}{\lambda_{0}(\zeta)}-1=\frac{\operatorname{\rm
e}^{-\mathsf{h}_{0}(\zeta)}-\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}}{1+\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}},\qquad\frac{\lambda_{0}(\zeta)}{\lambda_{\tau}(\zeta)}-1=-\frac{\operatorname{\rm
e}^{-\mathsf{h}_{0}(\zeta)}-\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}}{1+\operatorname{\rm
e}^{-\mathsf{h}_{0}(\zeta)}},$

are trivial, and because $\mathsf{h}_{0}$ and $\mathsf{h}_{\tau}$ are real-
valued along $\mathsf{\Sigma}_{2}=(-\infty,0)$, these equalities give

$\left|\frac{\lambda_{\tau}(\zeta)}{\lambda_{0}(\zeta)}-1\right|+\left|\frac{\lambda_{0}(\zeta)}{\lambda_{\tau}(\zeta)}-1\right|\leq
2\left|\operatorname{\rm e}^{-\mathsf{h}_{0}(\zeta)}-\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(\zeta)}\right|,\quad\zeta<0.$

For $|\zeta|\leq\tau^{\nu}$ we use Lemma 7.3 and estimate

$\left|\frac{\lambda_{\tau}(\zeta)}{\lambda_{0}(\zeta)}-1\right|+\left|\frac{\lambda_{0}(\zeta)}{\lambda_{\tau}(\zeta)}-1\right|=\mathcal{O}\left(\frac{\operatorname{\rm
e}^{-\mathsf{s}-\mathsf{t}|\zeta|}}{\tau^{1-2\nu}}\right),\quad\tau\to+\infty,\quad-\tau^{\nu}\leq\zeta\leq
0,$

whereas for $\zeta\leq-\tau^{\nu}$ we use instead the definition of
$\mathsf{h}_{0}$ in (4.6) and Definition 4.1–(iii) and write

$\left|\frac{\lambda_{\tau}(\zeta)}{\lambda_{0}(\zeta)}-1\right|+\left|\frac{\lambda_{0}(\zeta)}{\lambda_{\tau}(\zeta)}-1\right|\leq
2\operatorname{\rm e}^{-\mathsf{s}-(\mathsf{t}+\eta)|\zeta|}.$

We combine these two inequalities with Propositions 6.1 and 6.2, and apply
them to (7.11). As a result, we learn that there exist $M>0,\eta>0,\tau_{0}>0$
for which the estimate

$\left|\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\zeta)-\mathbf{I}\right|\leq
M|\zeta|^{1/2}\operatorname{\rm
e}^{-\mathsf{s}-\eta|\zeta|}\left(\frac{1}{\tau^{1-2\nu}}\chi_{(-\tau^{\nu},0]}(\zeta)+\chi_{(-\infty,-\tau^{\nu})}(\zeta)\right),\quad\zeta\leq
0,$ (7.12)

is valid for any $\mathsf{s}\geq-\mathsf{s}_{0},$
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$, $\tau\geq\tau_{0}$. After
possibly changing the values of $\eta,M$, the result follows from standard
arguments. ∎

Now that we controlled the asymptotic behavior for the jump matrix
$\mathbf{J}_{\mathbf{\Psi}_{\tau}}$, we are ready to obtain small norm
estimates for $\mathbf{\Psi}_{\tau}$ itself. We summarize these estimates in
the next result. For that, we recall the matrix norm notations introduced in
(6.1),(6.2),(6.3).

######  Theorem 7.7.

Fix $\mathsf{t}_{0}\in(0,1)$ and $\mathsf{s}_{0}>0$. There exists
$\tau_{0}=\tau_{0}(\mathsf{t}_{0},\mathsf{s}_{0})>0$ for which the solution
$\mathbf{\Psi}_{\tau}$ uniquely exists for any $\tau\geq\tau_{0}$ and any
$\mathsf{s}\geq-\mathsf{s}_{0},\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.
Furthermore, it satisfies the following asymptotic properties.

Its boundary value $\mathbf{\Psi}_{\tau,-}$ exists along $\mathsf{\Sigma}$,
and satisfies the estimate

$\|\mathbf{\Psi}_{\tau,-}-\mathbf{I}\|_{L^{2}(\mathsf{\Sigma})}=\mathcal{O}\left(\tau^{-\kappa}\right),\quad\tau\to+\infty,$

for any $\kappa\in(0,1)$, where the error term, for a given $\kappa$, is
uniform for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

For $\tau$ sufficiently large, the solution $\mathbf{\Psi}_{\tau}$ admits the
representation

$\mathbf{\Psi}_{\tau}(\zeta)=\mathbf{I}+\frac{1}{2\pi\mathrm{i}}\int_{\mathsf{\Sigma}}\frac{\mathbf{\Psi}_{\tau,-}(w)(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(w)-\mathbf{I})}{w-\zeta}\mathrm{d}w,\quad\zeta\in\mathbb{C}\setminus\mathsf{\Sigma}.$
(7.13)

Still for $\tau$ sufficiently large, $\mathbf{\Psi}_{\tau}$ satisfies

$\mathbf{\Psi}_{\tau}(\zeta)=\mathbf{I}+\mathbf{\Psi}_{\tau}^{(1)}\frac{1}{\zeta}+\mathcal{O}(\zeta^{-2}),\quad\zeta\to\infty,\quad\text{with}\quad\mathbf{\Psi}_{\tau}^{(1)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{1}{2\pi\mathrm{i}}\int_{\mathsf{\Sigma}}\mathbf{\Psi}_{\tau,-}(\xi)(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\xi)-\mathbf{I})\mathrm{d}\xi.$
(7.14)

###### Proof.

The small norm estimates provided by Lemmas 7.4, 7.5 and 7.6 allow us to apply
the small norm theory for Riemann-Hilbert problems (see for instance [34,
37]), and the claims follow with standard methods. We stress that for this
statement we only need the $L^{2}$ and $L^{\infty}$ estimates from the
aforementioned lemmas, but the $L^{1}$ estimates provided by them will be
useful later. ∎

###  7.2. Proof of main results of the section

We are ready to prove the main results of this section.

###### Proof of Theorem 7.1.

During the whole proof we identify $\nu=(1-\kappa)/2$.

The matrix ${\mathbf{\Phi}}_{0}$ always exists, whereas Theorem 7.7 provides
the existence of $\mathbf{\Psi}_{\tau}$ for $\tau$ sufficiently large. From
the relation (7.4) we obtain the claimed existence of $\mathbf{\Phi}_{\tau}$.

Comparing (7.6) with (7.14) we obtain the identity

$\mathbf{\Phi}_{\tau}^{(1)}={\mathbf{\Phi}}_{0}^{(1)}+\mathbf{\Psi}_{\tau}^{(1)}.$

Writing

$\mathbf{\Psi}_{\tau}^{(1)}=-\frac{1}{2\pi\mathrm{i}}\int_{\mathsf{\Sigma}}(\mathbf{\Psi}_{\tau,-}(\xi)-\mathbf{I})(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\xi)-\mathbf{I})\mathrm{d}\xi-\frac{1}{2\pi\mathrm{i}}\int_{\mathsf{\Sigma}}\mathbf{(}\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\xi)-\mathbf{I})\mathrm{d}\xi,$
(7.15)

and using Cauchy-Schwartz,

$|\mathbf{\Psi}_{\tau}^{(1)}|\leq\|\mathbf{\Psi}_{\tau,-}-\mathbf{I}\|_{L^{2}(\mathsf{\Sigma})}\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{2}(\mathsf{\Sigma})}+\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}(\mathsf{\Sigma})},$

and from Lemmas 7.4, 7.5, 7.6 and Theorem 7.7, the right-hand side above is
$\mathcal{O}(\tau^{-\kappa})$, for any $\kappa\in(0,1)$ and uniformly for
$\mathsf{s},\mathsf{t}$ as in (7.1), proving (7.2).

To prove the asymptotic formula (7.3) we follow arguments presented in [48,
Theorem 3.1] and [3, Lemma 2], with minor modifications to handle the
uniformity on the unbounded set $|x|\leq\tau^{-\nu}$ as claimed.

First off, the jump matrix $\mathbf{J}_{\mathbf{\Psi}_{\tau}}$ is $C^{\infty}$
on $\mathsf{\Sigma}$, in particular Hölder continuous, implying that
$\mathbf{\Psi}_{\tau}$ extends continuously to its boundary values
$\mathbf{\Psi}_{\tau,\pm}$. Accounting also for the behavior of
$\mathbf{\Psi}_{\tau}$ at $\infty$ and combining with the maximum principle,

$\|\mathbf{\Psi}_{\tau}\|_{L^{\infty}(\mathbb{C}\setminus\mathsf{\Sigma})}\leq
M_{\tau}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max\left\\{\|\mathbf{\Psi}_{\tau,+}\|_{L^{\infty}(\mathsf{\Sigma})},\|\mathbf{\Psi}_{\tau,-}\|_{L^{\infty}(\mathsf{\Sigma})}\right\\},$
(7.16)

where the constant $M_{\tau}$ is finite. For a point
$x\in\mathsf{\Sigma}\setminus\\{0\\}$ and $\varepsilon>0$, we consider the
arcs $C^{\pm}_{\varepsilon}(x)$ of the disk centered at $x$ and radius
$\varepsilon$ which are on the $\pm$ side of $\mathsf{\Sigma}$. We then set

$\mathsf{\Sigma}^{\pm}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(\mathsf{\Sigma}\setminus
D_{\varepsilon}(x)\right)\cup C^{\pm}_{\varepsilon}(x),$

with the orientation induced from $\mathsf{\Sigma}$. We deform contour in the
integral representation (9.4) and then send $z\to x$, obtaining that

$\mathbf{\Psi}_{\tau,\pm}(x)=\mathbf{I}+\frac{1}{2\pi\mathrm{i}}\int_{\mathsf{\Sigma}^{\mp}}\frac{\mathbf{\Psi}_{\tau,-}(s)(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(s)-\mathbf{I})}{s-x}\mathrm{d}s.$
(7.17)

From standard estimates and using (7.16), the just written equation yields

$|\mathbf{\Psi}_{\tau,\pm}(x)|\leq
1+\frac{1}{\pi\varepsilon}\|\mathbf{\Psi}_{\tau,-}\|_{L^{\infty}(\mathsf{\Sigma}^{\mp})}\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}(\mathsf{\Sigma}^{\mp})}\leq
1+\frac{1}{\pi\varepsilon}M_{\tau}\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}(\mathsf{\Sigma}^{\mp})},$

and therefore

$M_{\tau}\leq\left(1-\frac{1}{\pi\varepsilon}\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}(\mathsf{\Sigma}^{\mp})}\right)^{-1}.$
(7.18)

Lemmas 7.4, 7.5 and 7.6 provide $L^{p}$ estimates for
$\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}$ along $\mathsf{\Sigma}$.
Exploring that $\mathsf{\Sigma}^{\pm}$ is obtained from $\mathsf{\Sigma}$
after a small deformation around the point $x=\mathcal{O}(\tau^{-\nu})$, it is
straightforward to see that the same estimates hold in
$\mathsf{\Sigma}^{\pm}$, which can be summarized as

$|\mathbf{J}_{\mathbf{\Psi}_{\tau}}(\zeta)-\mathbf{I}|\leq M\operatorname{\rm
e}^{-\eta\min\\{|\zeta|,|\zeta|^{3/2}\\}}\left(\frac{1}{\tau^{\kappa}}\chi_{\\{|\zeta|\leq\tau^{\nu}\\}}(\zeta)+\chi_{\\{|\zeta|>\tau^{\nu}\\}}(\zeta)\right),\quad\zeta\in\mathsf{\Sigma}^{\pm},$
(7.19)

for some constants $\eta,M>0$ which may depend on
$\mathsf{s}_{0},\mathsf{t}_{0},\tau_{0}$ but are independent of
$\mathsf{s}\geq\mathsf{s}_{0},\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}],\tau\geq\tau_{0}$,
see (7.9), (7.10) and (7.12). Combining with (7.18), we conclude in particular
that $M_{\tau}\leq 2$, for $\mathsf{s},\mathsf{t},\tau$ in the same range of
values. Having in mind (7.4), this bound on $M_{\tau}$ applied to (7.17) is
enough to ensure that
$\mathbf{\Psi}_{\tau,\pm}-\mathbf{I}=\mathcal{O}(\tau^{-\nu})$, but to obtain
the decay in $x$ for the error claimed in (7.3) a little more care is needed
as follows.

First off, we split the integral in (7.17) into two, namely along

$J_{x}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{\zeta\in\mathsf{\Sigma}^{-}\mid|\zeta-x|\geq|x|/2\\}\quad\text{and}\quad\mathsf{\Sigma}^{-}\setminus
J_{x}.$

For the integral over $J_{x}$, we estimate as

$\left|\int_{J_{x}}\frac{\mathbf{\Psi}_{\tau,-}(s)(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(s)-\mathbf{I})}{s-x}\mathrm{d}s\right|\leq
2M_{\tau}\sup_{\zeta\in
J_{x}}\frac{1}{|\zeta-x|}\|\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}\|_{L^{1}(J_{x})}=\mathcal{O}(\tau^{-\kappa}|x|^{-1}),$

where we used (7.19), the fact that $x=\mathcal{O}(\tau^{-\kappa})$ and again
the bound $M_{\tau}\leq 2$. Observing that $|\zeta-x|\geq\varepsilon$ along
the remaining piece, a similar argument yields

$\left|\frac{\mathbf{\Psi}_{\tau,-}(s)(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(s)-\mathbf{I})}{s-x}\right|\leq\frac{4}{\varepsilon}\left|\mathbf{J}_{\mathbf{\Psi}_{\tau}}(s)-\mathbf{I}\right|,\quad
s\in\mathsf{\Sigma}^{-}\setminus J_{x},$

and again from (7.19) we see that the right-hand side above decays
exponentially in $x$ when $x\to\infty$ and is $\mathcal{O}(\tau^{-\kappa})$
when $\tau\to\infty$. From (7.17) we thus obtain

$\left|\mathbf{\Psi}_{\tau,+}(x)-\mathbf{I}\right|=\mathcal{O}\left(\frac{1}{(1+|x|)\tau^{\kappa}}\right),\quad\tau\to\infty,$

uniformly for $x\in\mathsf{\Sigma},|x|\leq\tau^{-\nu}$, and uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0},\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$. In virtue of (7.4), this proves (7.3). ∎

######  Remark 7.8.

For admissible functions $\mathsf{h}_{\tau}$, the asymptotics (4.3) as
$\zeta\to\infty$ of $\mathbf{\Phi}_{\tau}$ is valid uniformly in
$\tau,\mathsf{t}$ and $\mathsf{s}$, in the sense that for any
$\mathsf{t}_{0}\in(0,1)$ and any $\mathsf{s}_{0}>0$, there exist $K>0$ and
$R>0$ such that

$\left|\mathbf{\Phi}_{\tau}(\zeta)\operatorname{\rm
e}^{\frac{2}{3}\zeta^{2/3}\bm{\sigma}_{3}}\mathbf{U}_{0}\zeta^{\bm{\sigma}_{3}/4}-\mathbf{I}\right|\leq\frac{K}{|\zeta|},\quad\text{whenever
}|\zeta|\geq
R,\;\mathsf{s}\geq-\mathsf{s}_{0},\;\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0},$

and we emphasize that $K$ and $R$ are independent of $\mathsf{s},\mathsf{t}$.
To see that this is true, in virtue of (7.4) it is enough to show that the
asymptotics (5.10) and (7.6) are uniform in the same sense, we indicate the
proof for the latter and the former is analogous.

Using the trivial identity $1/(w-\zeta)=s/(s(s-\zeta))-1/\zeta$, we express
(7.13) as

$\mathbf{\Psi}_{\tau}(\zeta)=\mathbf{I}-\frac{1}{2\pi\mathrm{i}\zeta}\int_{\mathsf{\Sigma}}\mathbf{\Psi}_{\tau,-}(w)(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(w)-\mathbf{I})\mathrm{d}w+\frac{1}{2\pi\mathrm{i}\zeta}\int_{\mathsf{\Sigma}}\frac{w\mathbf{\Psi}_{\tau,-}(w)(\mathbf{J}_{\mathbf{\Psi}_{\tau}}(w)-\mathbf{I})}{w-\zeta}\mathrm{d}w.$

Because $\mathbf{\Psi}_{\tau,-}\in\mathbf{I}+L^{1}(\mathsf{\Sigma})$ and
$\mathbf{J}_{\mathbf{\Psi}_{\tau}}-\mathbf{I}$ decays pointwise exponentially
fast (and uniformly in $\mathsf{s},\mathsf{t}$ as claimed), the two integrals
can be bounded uniformly in $\zeta,\tau,\mathsf{s}$ as claimed, and the
uniform decay for $\mathbf{\Psi}_{\tau}$ as claimed follows.

To finish this section, it remains to prove Theorem 7.2, and for that end we
first establish a lemma.

######  Lemma 7.9.

Fix $\mathsf{s}\in\mathbb{R}$ and $\mathsf{t}_{0}\in(0,1)$. For any
$\nu\in(0,1/2)$, there exists $\eta=\eta(\nu)>0$ independent of
$\mathsf{s},\mathsf{t},\tau$ for which the estimates

$\int_{|x|\geq\tau^{\nu}}\frac{\operatorname{\rm
e}^{\mathsf{t}x-u}}{(1+\operatorname{\rm
e}^{\mathsf{t}x-u})^{2}}\left|\left[({\mathbf{\Delta}}_{0}(x\mid
u,\mathsf{t})^{-1}{\mathbf{\Phi}}_{0,+}(x\mid
u,\mathsf{t})^{-1}({\mathbf{\Phi}}_{0,+}{\mathbf{\Delta}}_{0})^{\prime}(x\mid
u,\mathsf{t})\right]_{21}\right|\mathrm{d}x=\mathcal{O}(\operatorname{\rm
e}^{-u-\eta\tau^{\nu}})$ (7.20)

and

$\int_{|x|\geq\tau^{\nu}}\frac{\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}}{\left(1+\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}\right)^{2}}\left|\left[\mathbf{\Delta}_{\tau}(x\mid
u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}\right|\mathrm{d}x=\mathcal{O}(\operatorname{\rm
e}^{-u-\eta\tau^{\nu}})$ (7.21)

are valid as $\tau\to\infty$, uniformly for $u\geq\mathsf{s}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$. In particular, both
integrands are integrable over $\mathbb{R}$.

###### Proof.

We prove (7.21) which is slightly more technical because the integrand depends
on $\tau$, the estimate (7.20) follows in a similar manner.

Having in mind Remark 7.8, we use the expansion (4.3) to estimate

$\mathbf{\Phi}_{\tau,+}(x)\mathbf{\Delta}_{\tau}(x)=\left(1+\mathcal{O}(x^{-1})\right)x^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\left(\mathbf{I}+\chi_{(-\infty,0)}(x)\operatorname{\rm
e}^{\frac{4}{3}x^{3/2}_{+}}(1-\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(x)})\mathbf{E}_{21}\right)\operatorname{\rm
e}^{-\frac{2}{3}x_{+}^{3/2}\bm{\sigma}_{3}},\quad x\in E_{\tau}.$

Observe that the factor $\chi_{(-\infty,0)}(x)\operatorname{\rm
e}^{\frac{4}{3}x^{3/2}_{+}}(1-\operatorname{\rm e}^{-\mathsf{h}_{\tau}(x)})$
is bounded, thanks to the facts that $x_{+}^{3/2}\in\mathrm{i}\mathbb{R}$ and
$\operatorname{Re}H>0$ for $x<0$ (see Definition 4.1–(iii)). The identity
above can be differentiated, and using it we obtain the crude bound

$\left[\mathbf{\Delta}_{\tau}(x\mid u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}=\operatorname{\rm
e}^{-\frac{4}{3}x_{+}^{3/2}}\mathcal{O}(|x|^{3}),\quad x\in E_{\tau},$

which is non-optimal in $x$, but will be enough for the coming estimates, and
which is valid uniformly for
$u\geq-\mathsf{s}_{0},\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$ as
$\tau\to\infty$. Thus, to conclude the result it is enough to estimate each of
the integrals

$I_{-}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{-\infty}^{-\tau^{\nu}}|x|^{3}\frac{\operatorname{\rm
e}^{\mathsf{h}_{\tau}(x\mid u)}}{(1+\operatorname{\rm
e}^{\mathsf{h}_{\tau}(x\mid u)})^{2}}\mathrm{d}x\quad\text{and}\quad
I_{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\tau^{\nu}}^{+\infty}|x|^{3}\frac{\operatorname{\rm
e}^{-\frac{4}{3}x^{3/2}}\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}}{(1+\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid u)})^{2}}\mathrm{d}x.$

To estimate $I_{-}$, we use the inequalities $v/(1+v)\leq 1$ and $1/(1+v)\leq
1/v$, valid for $v>0$, to estimate

$\frac{\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid u)}}{(1+\operatorname{\rm
e}^{\mathsf{h}_{\tau}(x\mid u)})^{2}}\leq\operatorname{\rm
e}^{-\mathsf{h}_{\tau}(x\mid u)},$

and using now the inequality from Definition 4.1–(iii) along
$\mathsf{\Sigma}_{2}=(-\infty,0)$, we obtain

$I_{-}\leq\operatorname{\rm
e}^{-u}\int_{-\infty}^{-\tau^{\nu}}|x|^{3}\operatorname{\rm
e}^{-\eta|x|}\mathrm{d}x=\mathcal{O}(\operatorname{\rm
e}^{-u-\widetilde{\eta}\tau^{\nu}/2})$

for some $\widetilde{\eta}>0$. In a similar manner, we also obtain that

$I_{+}\leq\operatorname{\rm
e}^{-u}\int_{\tau^{\nu}}^{+\infty}|x|^{3}\operatorname{\rm
e}^{-\frac{4}{3}x^{3/2}+\widehat{\eta}x^{2/3-\varepsilon}}\mathrm{d}x=\mathcal{O}(\operatorname{\rm
e}^{-u-\tau^{3\nu/2}}),$

where now for the last equality we used Definition 4.1–(iii) along
$\mathsf{\Sigma}_{0}=(0,\infty)$.

∎

###### Proof of Theorem 7.2.

As in the previous proof, we identify $\kappa\in(0,1)$ from Theorem 7.1 with
$\nu=(1-\kappa)/2\in(0,1/2)$. Thanks to (7.21),

$\int_{-a\tau}^{b\tau}\frac{\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}}{\left(1+\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}\right)^{2}}\left[\mathbf{\Delta}_{\tau}(x\mid
u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}\mathrm{d}x\\\
=\int_{-\tau^{\nu}}^{\tau^{\nu}}\frac{\operatorname{\rm
e}^{\mathsf{h}_{\tau}(x\mid u)}}{\left(1+\operatorname{\rm
e}^{\mathsf{h}_{\tau}(x\mid u)}\right)^{2}}\left[\mathbf{\Delta}_{\tau}(x\mid
u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}\mathrm{d}x+\mathcal{O}(\operatorname{\rm
e}^{-u-\eta\tau^{\nu}}),$

valid as $\tau\to\infty$ and uniformly for $u\geq\mathsf{s}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$. Next, we use Lemma 7.3
and (7.3) to ensure that

$\int_{-\tau^{\nu}}^{\tau^{\nu}}\frac{\operatorname{\rm
e}^{\mathsf{h}_{\tau}(x\mid u)}}{\left(1+\operatorname{\rm
e}^{\mathsf{h}_{\tau}(x\mid u)}\right)^{2}}\left[\mathbf{\Delta}_{\tau}(x\mid
u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}\mathrm{d}x=\\\
(1+\mathcal{O}(\tau^{-\kappa}))\int_{-\tau^{\nu}}^{\tau^{\nu}}\frac{\operatorname{\rm
e}^{\mathsf{h}_{0}(x)}}{(1+\operatorname{\rm
e}^{\mathsf{h}_{0}(x)})^{2}}\left[({\mathbf{\Delta}}_{0}(x\mid
u,\mathsf{t})^{-1}{\mathbf{\Phi}}_{0,+}(x\mid
u,\mathsf{t})^{-1}({\mathbf{\Phi}}_{0,+}{\mathbf{\Delta}}_{0})^{\prime}(x\mid
u,\mathsf{t})\right]_{21}\mathrm{d}x.$

With the help of the calculation

$\frac{\operatorname{\rm e}^{\mathsf{h}_{0}(x)}}{(1+\operatorname{\rm
e}^{\mathsf{h}_{0}(x)})^{2}}=\frac{1}{(1+\operatorname{\rm
e}^{\mathsf{h}_{0}(x)})(1+\operatorname{\rm
e}^{-\mathsf{h}_{0}(x)})}=\frac{\operatorname{\rm
e}^{-\mathsf{h}_{0}(x)}}{(1+\operatorname{\rm
e}^{-\mathsf{h}_{0}(x)})^{2}}=\frac{\operatorname{\rm
e}^{\mathsf{t}x-u}}{(1+\operatorname{\rm e}^{\mathsf{t}x-u})^{2}}$

we recognize the integrand from (7.20), and then conclude that

$\int_{-a\tau}^{b\tau}\frac{\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}}{\left(1+\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}\right)^{2}}\left[\mathbf{\Delta}_{\tau}(x\mid
u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}\mathrm{d}x=\\\
(1+\mathcal{O}(\tau^{-\kappa}))\int_{-\infty}^{\infty}\frac{\operatorname{\rm
e}^{\mathsf{t}x-u}}{(1+\operatorname{\rm
e}^{\mathsf{t}x-u})^{2}}\left[({\mathbf{\Delta}}_{0}(x\mid
u,\mathsf{t})^{-1}{\mathbf{\Phi}}_{0,+}(x\mid
u,\mathsf{t})^{-1}({\mathbf{\Phi}}_{0,+}{\mathbf{\Delta}}_{0})^{\prime}(x\mid
u,\mathsf{t})\right]_{21}\mathrm{d}x+\mathcal{O}(\operatorname{\rm
e}^{-u-\eta\tau^{\nu}}).$

Finally, with arguments very similar to the ones used in the proof of Lemma
7.9, we see that this remaining integral on the right-hand side is
$\mathcal{O}(\operatorname{\rm e}^{-u})$. Everything combined, we just proved
that

$\int_{-\tau a}^{\tau b}\frac{\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}}{\left(1+\operatorname{\rm e}^{\mathsf{h}_{\tau}(x\mid
u)}\right)^{2}}\left[\mathbf{\Delta}_{\tau}(x\mid
u)^{-1}\mathbf{\Phi}_{\tau,+}(x\mid
u)^{-1}\left(\mathbf{\Phi}_{\tau,+}\mathbf{\Delta}_{\tau}\right)^{\prime}(x\mid
u)\right]_{21}\mathrm{d}x=\\\ \int_{-\infty}^{\infty}\frac{\operatorname{\rm
e}^{\mathsf{t}x-u}}{(1+\operatorname{\rm
e}^{\mathsf{t}x-u})^{2}}\left[({\mathbf{\Phi}}_{0,+}(x\mid
u,\mathsf{t}){\mathbf{\Delta}}_{0}(x\mid
u,\mathsf{t}))^{-1}({\mathbf{\Phi}}_{0,+}{\mathbf{\Delta}}_{0})^{\prime}(x\mid
u,\mathsf{t})\right]_{21}\mathrm{d}x+\mathcal{O}(\tau^{-\kappa}\operatorname{\rm
e}^{-u}),\quad\tau\to+\infty.$

We now integrate in $u\in[-\mathsf{s},+\infty)$ and use Equation (5.8) and the
limit

$\lim_{s\to-\infty}\mathsf{L}^{\kern-1.04996pt\operatorname{Ai}}(s,T)=1,$

which is valid by dominated convergence since $|1+\operatorname{\rm
e}^{\alpha}|^{-1}\leq 1$ for every $\alpha\in\mathbb{R}$, to conclude the
proof. ∎

##  8\. The underlying equilibrium measure and related quantities

As we mentioned earlier, one of the main objects we will analyze is the RHP
for the orthogonal polynomials associated to (2.3)–(2.4). In this analysis,
the model problem we discussed in the last few sections plays a central role,
and another important quantity is the associated equilibrium measure and
related objects that we discuss in this section.

###  8.1. The equilibrium measure

A major role in our calculations is played by the equilibrium measure for the
polynomial potential $V$, which is the unique probability measure $\mu_{V}$ on
$\mathbb{R}$ for which the quantity

$\iint\log\frac{1}{|x-y|}\mathrm{d}\mu(x)\mathrm{d}\mu(y)+\int
V(x)\mathrm{d}\mu(x)$

attains its minimum over all Borel probability measures $\mu$ supported on
$\mathbb{R}$. Its existence and uniqueness is assured by standard results, see
for instance [56], and its regularity that we now discuss is of particular
relevance.

The measure $\mu_{V}$ is supported on a finite union of bounded intervals and
is absolutely continuous with respect to the Lebesgue measure [36]. Following
Assumption 2.1–(i), we assume that $\mu_{V}$ is one-cut, that is, it is
supported on a single interval that we take to be of the form

$\operatorname{supp}\mu_{V}=[-a,0],\quad a>0,$

and regular, meaning that the density of $\mu_{V}$ vanishes as a square-root
at a neighborhood of the endpoints, does not vanish on $(-a,0)$, and the
Euler-Lagrange equations are valid with strict inequality outside the support,

$\int\log\frac{1}{|x-y|}\mathrm{d}\mu_{V}(y)+\frac{1}{2}V(x)+\ell_{V}\;\begin{cases}>0,&x\in\mathbb{R}\setminus\operatorname{supp}\mu_{V},\\\
=0,&x\in\operatorname{supp}\mu_{V},\end{cases}$ (8.1)

for some constant $\ell_{V}\in\mathbb{R}$. The notions just introduced are
consistent with the notions and notations that we already introduced and used
in Section 2.

A transformation of the equilibrium measure of particular interest is its
Cauchy transform,

$C^{\mu_{V}}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int\frac{\mathrm{d}\mu_{V}(x)}{x-z},\quad
z\in\mathbb{C}\setminus\operatorname{supp}\mu_{V}.$

Using the Euler-Lagrange identity, it can be shown that $C^{\mu_{V}}$
satisfies an algebraic equation of the form

$\left(C^{\mu_{V}}(z)+\frac{V^{\prime}(z)}{2}\right)^{2}=\frac{1}{4}z(z+a)h_{V}(z)^{2},\quad
z\in\mathbb{C}\setminus\operatorname{supp}\mu_{V},$ (8.2)

for some polynomial $h_{V}$ which does not vanish on $[-a,0]$ [50, 52].

We also associate to the equilibrium measure its $\phi$ function

$\phi(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{0}^{z}\left(C^{\mu_{V}}(s)+\frac{1}{2}V^{\prime}(s)\right)\mathrm{d}s,\quad
z\in\mathbb{C}\setminus(-\infty,0].$ (8.3)

The next result summarizes some properties of $\phi$ that will be needed
later.

######  Proposition 8.1.

The function $\phi$ has the following properties.

  1. (i)

The function $\phi$ is analytic on $\mathbb{C}\setminus(-\infty,0]$.

  2. (ii)

For $x\in(-a,0)$,

$\phi_{+}(x)+\phi_{-}(x)=0,\quad\text{and}\quad\phi_{+}(x)-\phi_{-}(x)=-2\pi\mathrm{i}\mu_{V}((x,0)).$

  3. (iii)

For $x\in(-\infty,-a)$,

$\phi_{+}(x)-\phi_{-}(x)=-2\pi\mathrm{i}.$

  4. (iv)

For $x\in\mathbb{R}\setminus[-a,0]$,

$\operatorname{Re}\phi_{+}(x)=\operatorname{Re}\phi_{-}(x)>0.$

  5. (v)

As $z\to\infty$ and some constant $\phi_{\infty}$,

$\phi(z)=\frac{V(z)}{2}+\ell_{V}-\log
z+\frac{\phi_{\infty}}{z}+\mathcal{O}(z^{-1}),$

where $\ell_{V}$ is as in (8.1).

  6. (vi)

The function $\phi$ satisfies the estimate

$\phi(z)=\frac{1}{3}h_{V}(0)a^{1/2}z^{3/2}(1+\mathcal{O}(z)),\quad z\to 0.$
(8.4)

###### Proof.

The proof is standard using the properties of the equilibrium measure, see for
instance [33]. ∎

###  8.2. The conformal map

Finally, using $\phi$ we construct a conformal map $\psi$, introduced formally
with the next result.

######  Proposition 8.2.

The function

$\psi(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(\frac{3}{2}\phi(z)\right)^{2/3},$

is a conformal map from a neighborhood of the origin to a disk
$D_{2\delta}(0)$, with
$U_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\psi^{-1}(D_{\delta}(0))$,