and admits an expansion of the form

$\psi(z)=\mathsf{c}_{V}z(1+\mathcal{O}(z)),\quad z\to
0,\qquad\mathsf{c}_{V}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=2^{-2/3}h_{V}(0)^{2/3}a^{1/3}>0.$
(8.5)

###### Proof.

The proof is also standard, and follows essentially from Proposition 8.1-(iv).
We omit the details. ∎

In the previous proposition, the factor $2\delta$ instead of $\delta$ is
chosen just for later convenience, for the statement of Proposition 8.3. Later
on, we use $\psi$ only over the smaller neighborhood $D_{\delta}(0)$.

As it is customary in RHP analysis, at a later stage we will need to glue the
model problem as a local parametrix for the original RHP for orthogonal
polynomials. This gluing procedure is done, in our case, using the conformal
map $\psi$. In usual situations, the jump matrices of the model local problem
are piecewise constant or yet homogeneous, and as such this procedure of using
the conformal map does not significantly alter them. However, in our situation
the jump involves the function $Q$, and consequently the jump will be altered
by the conformal map in a nontrivial way.

With the next result we introduce the necessary quantities needed to keep
track of this transformation. Recall the half rays $\mathsf{\Sigma}_{j}$,
$j=0,1,2,3,$ which were introduced in (4.1). For the next statement, we talk
about neighborhoods of $\mathsf{\Sigma}_{j}$, by which we mean open connected
sets that contain $\mathsf{\Sigma}_{j}\setminus\\{0\\}$ in their interior.

######  Proposition 8.3.

There exist neighborhoods $\mathcal{S}_{j}$ of $\mathsf{\Sigma}_{j}$ and a
function

$\mathsf{H}_{Q}:\mathcal{S}\to\mathbb{C},\quad\mathcal{S}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\bigcup_{j=0}^{3}\mathcal{S}_{j}$

with the following properties.

  1. (i)

For the value $\delta>0$ in Proposition 8.2, the inclusions

$D_{\delta}(0)\subset\mathcal{S}\quad\text{and}\quad\mathcal{S}_{j}\cap\mathcal{S}_{k}\subset
D_{\delta}(0),$

hold true for any $j\neq k$.

  2. (ii)

The function $\mathsf{H}_{Q}$ is $C^{\infty}$ on $\mathcal{S}$, and is an
extension of $Q\circ\psi^{-1}$ from $D_{\delta}(0)$, that is,

$\mathsf{H}_{Q}(w)=Q(\psi^{-1}(w)),\quad|w|<\delta.$ (8.6)

  3. (iii)

The function $\mathsf{H}_{Q}$ is analytic on $D_{\delta}(0)$, extends
continuously up to the boundary of $D_{\delta}(0)$ and satisfies

$\mathsf{H}_{Q}(w)=-\mathsf{c}_{\mathsf{H}}w+\mathcal{O}(w),\quad\mathsf{c}_{\mathsf{H}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\mathsf{t}}{\mathsf{c}_{V}},$
(8.7)

uniformly for $|w|\leq\delta$, where we recall that $\mathsf{t}$ and
$\mathsf{c}_{V}$ are as in (2.7) and (8.5), respectively.

  4. (iv)

For some constants $\widehat{\eta}>\eta>0$, the function $\mathsf{H}_{Q}$
satisfies the estimates

$-\widehat{\eta}|w|\leq\operatorname{Re}\mathsf{H}_{Q}(w)\leq-\eta|w|\qquad\text{for
every }w\in\mathcal{S}_{0}\setminus D_{\delta}(0),$

and

$\operatorname{Re}\mathsf{H}_{Q}(w)\geq\eta|w|\qquad\text{for every
}w\in\left(\mathcal{S}_{1}\cup\mathcal{S}_{2}\cup\mathcal{S}_{3}\right)\setminus
D_{\delta}(0).$

###### Proof.

We construct the set $\mathcal{S}_{j}$ as tubular neighborhoods of
$\mathsf{\Sigma}_{j}$ away from the origin, and as disks near the origin,
namely

$\mathcal{S}_{j}=\mathcal{S}_{j}(\delta^{\prime})=\left\\{w\in\mathbb{C}\mid\inf_{w^{\prime}\in\mathsf{\Sigma}_{j}}|w-w^{\prime}|<\delta^{\prime}\right\\}\cup
D_{\delta}(0).$

By choosing $\delta^{\prime}>0$ sufficiently small, in particular smaller than
$\delta>0$, property (i) is immediate.

The function

$\mathsf{H}_{Q}(w)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=Q(\psi^{-1}(w)),\quad|w|<\delta,$

is obviously analytic on $D_{\delta}(0)$, satisfies (iii) and admits an
extension to the larger open set $D_{2\delta}(0)$. A standard argument using
partitions of unity allows us to extend it to the sets $\mathcal{S}_{j}$’s as
claimed by (ii), also making sure that it satisfies (iv). ∎

##  9\. Associated orthogonal polynomials

The first and arguably major step towards understanding
$\mathsf{L}^{Q}_{n}(\mathsf{s})$ is to study several quantities related to the
orthogonal polynomials for the varying weight (2.17), as we introduce next.

###  9.1. Orthogonal polynomials and related quantities

Denote by $\mathsf{P}_{k}=\mathsf{P}_{k}^{(n,\mathsf{s})}$ the monic
orthogonal polynomial of degree $k$ for the weight $\omega_{n}$ in (2.17),

$\mathsf{P}_{k}^{(n,\mathsf{s})}(x)=x^{k}+\text{(lower degree
terms)},\qquad\int_{\mathbb{R}}\mathsf{P}_{k}^{(n,\mathsf{s})}(x)x^{j}\omega_{n}(x)\mathrm{d}x=0,\quad
j=0,\ldots,k-1.$ (9.1)

These polynomials depend on $\omega_{n}$, so ultimately also on $Q$, but we
refrain from stressing this dependence in the notation. We also denote by
$\upgamma_{k}^{(n,Q)}=\upgamma_{k}^{(n,Q)}(\mathsf{s})>0$ the corresponding
norming constant, determined by

$\frac{1}{\upgamma_{k}^{(n)}(\mathsf{s})^{2}}=\int_{\mathbb{R}}\mathsf{P}_{k}^{(n,\mathsf{s})}(x)^{2}\omega_{n}(x)\mathrm{d}x.$
(9.2)

We associate to the orthogonal polynomials their Christoffel-Darboux kernel,

$\mathsf{K}^{Q}_{n}(x,y)=\mathsf{K}^{Q}_{n}(x,y\mid\mathsf{s})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{k=0}^{n-1}\upgamma_{k}^{(n)}(\mathsf{s})^{2}\mathsf{P}_{k}^{(n,\mathsf{s})}(x)\mathsf{P}_{k}^{(n,\mathsf{s})}(y),$
(9.3)

stressing that we are not including the weight $\omega_{n}$ in this
definition. In particular, the identity

$\int_{-\infty}^{\infty}\mathsf{K}_{n}(x,x\mid\mathsf{s})\omega_{n}(x\mid\mathsf{s})\mathrm{d}x=n$
(9.4)

holds true for any $\mathsf{s}\in\mathbb{R}$ and follows immediately from
(9.2).

In a similar manner, we introduce the related quantities for the undeformed
weight $\operatorname{\rm e}^{-nV}$. The partition function $Z_{n}$ already
appeared in (2.2), the orthogonal polynomials $P_{k}=P_{k}^{(n)}$ are
determined by

$P_{k}^{(n)}(x)=x^{k}+\text{(lower degree
terms)},\qquad\int_{\mathbb{R}}P_{k}^{(n)}(x)x^{j}\operatorname{\rm
e}^{-nV(x)}\mathrm{d}x=0,\quad j=0,\ldots,k-1,$

and the norming constants and Christoffel-Darboux kernel are determined from

$\frac{1}{(\gamma_{k}^{(n)})^{2}}=\int_{\mathbb{R}}P_{k}^{(n)}(x)^{2}\operatorname{\rm
e}^{-nV(x)}\mathrm{d}x,\quad
K_{n}(x,y)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{k=0}^{n-1}(\gamma_{k}^{(n)})^{2}P_{k}^{(n)}(x)P_{k}^{(n)}(y).$

The orthogonal polynomials $\mathsf{P}_{k}^{(n,\mathsf{s})}$ vary continuously
with $\mathsf{s}$, which is a consequence of Heine’s formula [33, Equation
(3.10)]. In particular, when taking the limit $\mathsf{s}\to+\infty$ we have
that $x^{k}\omega_{n}(x)\to x^{k}\operatorname{\rm e}^{-nV}$ both uniformly in
compacts and also in $L^{1}$, and
$|x|^{k}\omega_{n}(x)\leq|x|^{k}\operatorname{\rm e}^{-nV(x)}$. Thus,
dominated convergence then gives that all the just introduced undeformed
quantities are recovered from their deformed versions in the limit
$s\to+\infty$. This means, for instance, that the Christoffel-Darboux kernel
$K_{n}$ and the partition function $Z_{n}$ are recovered via

$K_{n}(x,y)=\mathsf{K}_{n}(x,y\mid\mathsf{s}=+\infty)\quad\text{and}\quad
Z_{n}=\mathsf{Z}_{n}(\mathsf{s}=+\infty).$ (9.5)

The next result will be key into transforming asymptotics for the orthogonal
polynomials to asymptotics for $\mathsf{L}^{Q}_{n}(s)$ itself.

######  Proposition 9.1.

The identity

$\log\mathsf{L}_{n}^{Q}(\mathsf{s})=-\int_{\mathsf{s}}^{\infty}\int_{-\infty}^{\infty}\mathsf{K}^{Q}_{n}(x,x\mid
u)\frac{\omega_{n}(x\mid u)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x\;\mathrm{d}u$ (9.6)

holds true for every $\mathsf{s}\in\mathbb{R}$.

######  Remark 9.2.

While we were finishing this manuscript, the work [29] was posted to the
ArXiv. Therein, they also derive the formula (9.6) in more general terms,
using the underlying RHP for IIKS-type integrable operators, see the first
displayed formula in page 28 therein. Similar formulas play a fundamental role
in the recent works [24, 25], see for instance (5.6) above. Our proof of (9.6)
relies solely on orthogonality properties, so we decided to present it
nevertheless.

###### Proof.

The equality

$\mathsf{Z}^{Q}_{n}(\mathsf{s})=n!\prod_{k=0}^{n-1}\upgamma_{k}^{(n)}(\mathsf{s})^{-2}$

is standard in random matrix theory. From this identity, (9.2) and the
orthogonality relations we derive the deformation formula

$\partial_{\mathsf{s}}\log\mathsf{Z}^{Q}_{n}(\mathsf{s})=-\int_{\mathbb{R}}\partial_{\mathsf{s}}\mathsf{K}^{Q}_{n}(x,x\mid\mathsf{s})\omega_{n}(x\mid\mathsf{s})\mathrm{d}x,$

which in fact is valid for general weights depending on an additional
parameter $\mathsf{s}$. To our knowledge, this last identity was first
observed by Krasovsky [47, Equation (14)]. We fix constants $L>\mathsf{s}>0$
and integrate the identity above,

$\log\mathsf{Z}^{Q}_{n}(\mathsf{s})=\log\mathsf{Z}^{Q}_{n}(L)+\int_{\mathsf{s}}^{L}\int_{-\infty}^{\infty}\mathsf{\partial}_{\mathsf{s}}\mathsf{K}^{Q}_{n}(x,x\mid
u)\omega_{n}(x\mid u)\mathrm{d}x\mathrm{d}u.$ (9.7)

We want to interchange the order of integration in the above. The derivative
$\partial_{\mathsf{s}}\mathsf{K}^{Q}_{n}$ is a polynomial of degree at most
$2n-2$ in $x$, and from Heine’s formula for orthogonal polynomials we see that
the polynomial coefficients of $\mathsf{K}_{n}$ are continuous functions of
$\mathsf{s}$. Therefore, for given $\mathsf{s},L$ there exists a constant
$M=M(\mathsf{s},L)>0$ for which the pointwise bound

$\left|\partial_{\mathsf{s}}\mathsf{K}^{Q}_{n}(x,x\mid u)\right|\leq
M\sup_{0\leq k\leq 2n-2}|x|^{k},\quad x\in\mathbb{R},$

is valid for every $u\in[\mathsf{s},L]$. Together with the inequalities
$0\leq\omega_{n}(x)\leq\operatorname{\rm e}^{-nV(x)}$, this bound ensures that
we can interchange order of integration in (9.7). After integration by parts,
we then obtain

$\log\mathsf{Z}^{Q}_{n}(\mathsf{s})=\log\mathsf{Z}^{Q}_{n}(L)+\int_{-\infty}^{\infty}\mathsf{K}^{Q}_{n}(x,x\mid
L)\omega_{n}(x\mid
L)\mathrm{d}x-\int_{-\infty}^{\infty}\mathsf{K}^{Q}_{n}(x,x\mid\mathsf{s})\omega_{n}(x\mid\mathsf{s})\mathrm{d}x\\\
-\int_{-\infty}^{\infty}\int_{\mathsf{s}}^{L}\mathsf{K}^{Q}_{n}(x,x\mid
u)\frac{\omega_{n}(x\mid u)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}u$

From the identity (9.4) the two single integrals cancel one another. The
integrand of the double integral is positive, so by Tonelli’s Theorem we can
interchange order of integration. After this interchange, we take the limit
$L\to+\infty$ and use (9.5) and (2.3) to conclude the proof. ∎

###  9.2. The Riemann-Hilbert Problem for orthogonal polynomials

We are ready to introduce the RHP for orthogonal polynomials for the weight
$\omega_{n}$ in (2.17). During this section, we keep using the matrix notation
that was already used in previous sections, recall for instance (3.1),(3.2)
and (3.3).

The RHP for orthogonal polynomials for the weight (2.17) asks for finding a
$2\times 2$ matrix-valued function $\mathbf{Y}$ with the following properties.

  1. Y-1.

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

  2. Y-2.

The function $\mathbf{Y}$ has continuous boundary values

$\mathbf{Y}_{\pm}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{\varepsilon\searrow
0}\mathbf{Y}(x\pm\mathrm{i}\varepsilon),\quad x\in\mathbb{R},$

which are related by the jump condition
$\mathbf{Y}_{+}(x)=\mathbf{Y}_{-}(x)\mathbf{J}_{\mathbf{Y}}(x)$,
$x\in\mathbb{R}$, with

$\mathbf{J}_{\mathbf{Y}}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{I}+\omega_{n}(x)\mathbf{E}_{12}.$

  3. Y-3.

As $z\to\infty$,

$\mathbf{Y}(z)=\left(\mathbf{I}+\mathcal{O}(z^{-1})\right)z^{n\bm{\sigma}_{3}}.$

Observe that $\mathbf{Y}=\mathbf{Y}^{(n)}(\cdot\mid\mathsf{s},Q)$ depends on
the index $n$ and also on $\mathsf{s}$ and $Q$, although we do not make this
dependence explicit in our notation. As shown by Fokas, Its and Kitaev [42],
for each $n$ the RHP above has a unique solution, which is explicitly given by

$\mathbf{Y}(z)=\begin{pmatrix}\mathsf{P}^{(n,\mathsf{s})}_{n}(z)&\displaystyle{\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}}\frac{P_{n}^{(n,\mathsf{s})}(x)}{x-z}\omega_{n}(x)\mathrm{d}x}\\\\[10.0pt]
-2\pi\mathrm{i}\upgamma^{(n,Q)}_{n-1}(\mathsf{s})^{2}\mathsf{P}^{(n,\mathsf{s})}_{n-1}(z)&\displaystyle{-\upgamma^{(n,Q)}_{n-1}(\mathsf{s})^{2}\dfrac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}}\frac{\mathsf{P}^{(n,\mathsf{s})}_{n-1}(x)}{x-z}\omega_{n}(x)\mathrm{d}x}\end{pmatrix},$

where $\upgamma_{k}^{(n,Q)}(\mathsf{s})$ and $\mathsf{P}_{k}^{(n,\mathsf{s})}$
are as in (9.1) and (9.2).

In particular, from this identity we obtain the relation

$\upgamma_{n-1}^{(n,Q)}(\mathsf{s})^{2}=-\frac{1}{2\pi\mathrm{i}}\left(\mathbf{Y}^{(n,1)}\right)_{21},$
(9.8)

where $\mathbf{Y}^{(n,1)}=\mathbf{Y}^{(n,1)}(\mathsf{s},Q)$ is the matrix
determined from the more detailed expansion

$\mathbf{Y}(z)=\mathbf{Y}^{(n)}(z)=\left(\mathbf{I}+\frac{1}{z}\mathbf{Y}^{(n,1)}+\frac{1}{z^{2}}\mathbf{Y}^{(n,2)}+\mathcal{O}(z^{-3})\right)z^{n\bm{\sigma}_{3}},\quad
z\to\infty.$ (9.9)

Also, the Christoffel-Darboux kernel (9.3) can be recast directly from
$\mathbf{Y}$ from the identity

$\mathsf{K}^{Q}_{n}(x,y\mid\mathsf{s})=\frac{1}{2\pi\mathrm{i}}\frac{1}{x-y}\mathbf{e}_{2}^{\mathrm{T}}\mathbf{Y}_{+}(y)^{-1}\mathbf{Y}_{+}(x)\mathbf{e}_{1},\quad
x,y\in\mathbb{R},\;x\neq y.$ (9.10)

In the confluent limit $x=y$, this formula yields

$\mathsf{K}^{Q}_{n}(x,x\mid\mathsf{s})=\frac{1}{2\pi\mathrm{i}}\mathbf{e}_{2}^{\mathrm{T}}\mathbf{Y}_{+}(x)^{-1}\mathbf{Y}^{\prime}_{+}(x)\mathbf{e}_{1},\quad
x\in\mathbb{R}.$ (9.11)

The remainder of this paper is dedicated to applying the Deift-Zhou method for
this RHP and collecting its consequences, analysis which will ultimately lead
to the proofs of our main results.

##  10\. The RHP analysis for the orthogonal polynomials

With all the preliminary work completed, we are finally at the stage of
performing the asymptotic analysis for the RHP-$\mathbf{Y}$ for orthogonal
polynomials that was introduced in Section 9. Most of the transformations are
standard, so we go over them quickly and without much detail. Care will be
taken in the construction of the parametrices, which are the steps where the
introduction of the factor $\sigma_{n}$ plays a major role. We also remind the
reader that the function $V$ and $Q$ are always assumed to satisfy Assumptions
2.1.

The function $\sigma_{n}=\sigma_{n}(z\mid\mathsf{s},Q)$ depends on
$\mathsf{s}\in\mathbb{R}$ and $Q$, and as such in all the steps below several
quantities will also depend on these parameters. Nevertheless, in most of the
work that follows the parameter $\mathsf{s}$ and the function $Q$ do not play
a major role so we omit them in our notations unless when needed to avoid
confusion.

###  10.1. First transformation: normalization at infinity

Recall the function $\phi$ introduced in (8.3). The first transformation,
which has the effect of normalizing the RHP as $z\to\infty$, takes the form

$\mathbf{T}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\operatorname{\rm
e}^{-n\ell_{V}\bm{\sigma}_{3}}\mathbf{Y}(z)\operatorname{\rm
e}^{n\left(\phi(z)-\frac{1}{2}V(z)\right)\bm{\sigma}_{3}},\quad
z\in\mathbb{C}\setminus\mathbb{R}.$ (10.1)

From the RHP for $\mathbf{Y}$ and the properties from Proposition 8.1, we
obtain that $\mathbf{T}$ satisfies the following RHP.

  1. T-1.

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

  2. T-2.

For $z\in\mathbb{R}$, it satisfies the jump
$\mathbf{T}_{+}(z)=\mathbf{T}_{-}(z)\mathbf{J}_{\mathbf{T}}(z)$, with

$\mathbf{J}_{\mathbf{T}}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{pmatrix}\operatorname{\rm
e}^{n(\phi_{+}(z)-\phi_{-}(z))}&\sigma_{n}(z)\operatorname{\rm
e}^{-n(\phi_{+}(z)+\phi_{-}(z))}\\\ 0&\operatorname{\rm
e}^{-n(\phi_{+}(z)-\phi_{-}(z))}\end{pmatrix},\quad z\in\mathbb{R}.$

  3. T-3.

As $z\to\infty$,

$\mathbf{T}(z)=\mathbf{I}+\frac{1}{z}\mathbf{T}_{1}+\mathcal{O}(z^{-2}),$

where the coefficient $\mathbf{T}_{1}$ is

$\mathbf{T}_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\operatorname{\rm
e}^{-n\ell_{V}\bm{\sigma}_{3}}\mathbf{Y}^{(n,1)}\operatorname{\rm
e}^{n\ell_{V}\bm{\sigma}_{3}}+\phi_{\infty}\bm{\sigma}_{3},$

and we recall that $\mathbf{Y}^{(n,1)}$ and $\phi_{\infty}$ were introduced in
(9.9) and in Proposition 8.1–(v), respectively.

From the properties (ii) of Proposition 8.1, the jump matrix for $\mathbf{T}$
simplifies in convenient ways. For $-a<z<0$,

$\displaystyle\mathbf{J}_{\mathbf{T}}(z)$
$\displaystyle=\begin{pmatrix}\operatorname{\rm
e}^{2n\phi_{+}(z)}&\sigma_{n}(z)\\\ 0&\operatorname{\rm
e}^{-2n\phi_{+}(z)}\end{pmatrix}$
$\displaystyle=\left(\mathbf{I}+\frac{1}{\sigma_{n}(z)}\operatorname{\rm
e}^{-2n\phi_{+}(z)}\mathbf{E}_{21}\right)\left(\sigma_{n}(z)\mathbf{E}_{12}-\frac{1}{\sigma_{n}(z)}\mathbf{E}_{21}\right)\left(\mathbf{I}+\frac{1}{\sigma_{n}(z)}\operatorname{\rm
e}^{2n\phi_{+}(z)}\mathbf{E}_{21}\right)$
$\displaystyle=\left(\mathbf{I}+\frac{1}{\sigma_{n}(z)}\operatorname{\rm
e}^{2n\phi_{-}(z)}\mathbf{E}_{21}\right)\left(\sigma_{n}(z)\mathbf{E}_{12}-\frac{1}{\sigma_{n}(z)}\mathbf{E}_{21}\right)\left(\mathbf{I}+\frac{1}{\sigma_{n}(z)}\operatorname{\rm
e}^{2n\phi_{+}(z)}\mathbf{E}_{21}\right),$

and for $z\in\mathbb{R}\setminus[-a,0]$,

$\displaystyle\mathbf{J}_{\mathbf{T}}(z)=\mathbf{I}+\sigma_{n}(z)\operatorname{\rm
e}^{-2n\phi_{+}(z)}\mathbf{E}_{12}.$

###  10.2. Second transformation: opening of lenses

From the identities just written for $\mathbf{J}_{\mathbf{T}}$ and Proposition
8.1-(ii), it follows that the diagonal entries of $\mathbf{J}_{\mathbf{T}}$
are highly oscillatory on $(-a,0)$ as $n\to\infty$. In the second
transformation of the RHP we perform the so-called opening of lenses, which
has the effect of moving this oscillatory behavior to a region where it
becomes exponentially decaying.

Define regions $\mathcal{G}^{\pm}$ on the $\pm$-side of $(-a,0)$ (the lenses,
see Figure 3), assuming in addition that for $U_{0}$ as in Proposition 8.2
these regions satisfy

$\psi(\partial\mathcal{G}^{\pm}\cap U_{0})\subset(0,\operatorname{\rm e}^{\pm
2\pi\mathrm{i}/3}\infty)\cup(0,\infty),$ (10.2)

which can always be achieved because $\psi:$ is conformal from a neighborhood
of $U_{0}$ to a neighborhood of $D_{\delta}(0)$.

The function $\sigma_{n}$ has no zeros and may have singularities, but these
are all poles due to the analyticity of $Q$ in a neighborhood of the real
axis. Therefore, the fraction $1/\sigma_{n}$ is analytic on a neighborhood of
the real axis. We use this fraction to transform

$\mathbf{S}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\mathbf{T}(z)\left(\mathbf{I}\mp\frac{1}{\sigma_{n}(z)}\operatorname{\rm
e}^{2n\phi(z)}\mathbf{E}_{21}\right),&z\in\mathcal{G}^{\pm},\\\
\mathbf{T}(z),&\text{elsewhere}.\end{dcases}$

$-a$$0$$\mathcal{G}^{+}$$\mathcal{G}^{-}$ Figure 3. The regions used for the
opening of lenses in the transformation $\mathbf{T}\mapsto\mathbf{S}$.

With

$\Gamma_{\mathbf{S}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{R}\cup\partial\mathcal{G}^{+}\cup\partial\mathcal{G}^{-},$

and using the jump properties of $\phi$ listed in Proposition 8.1, the matrix
$\mathbf{S}$ satisfies the following RHP.

  1. S-1.

The matrix
$\mathbf{S}:\mathbb{C}\setminus\Gamma_{\mathbf{S}}\to\mathbb{C}^{2\times 2}$
is analytic.

  2. S-2.

For $z\in\Gamma_{\mathbf{S}}$, it satisfies the jump
$\mathbf{S}_{+}(z)=\mathbf{S}_{-}(z)\mathbf{J}_{\mathbf{S}}(z)$, with

$\mathbf{J}_{\mathbf{S}}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\sigma_{n}(z)\mathbf{E}_{12}-\frac{1}{\sigma_{n}(z)}\mathbf{E}_{21},&-a<z<0,\\\
\mathbf{I}+\frac{1}{\sigma_{n}(z)}\operatorname{\rm
e}^{2n\phi(z)}\mathbf{E}_{21},&z\in\partial\mathcal{G}^{\pm}\setminus\mathbb{R},\\\
\mathbf{I}+\sigma_{n}(z)\operatorname{\rm
e}^{-2n\phi_{+}(z)}\mathbf{E}_{12},&z\in\mathbb{R}\setminus(-a,0).\end{dcases}$

  3. S-3.

As $z\to\infty$,

$\mathbf{S}(z)=\mathbf{I}+\frac{\mathbf{S}_{1}}{z}+\mathcal{O}(z^{-2}),\qquad\text{with}\quad\mathbf{S}_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{T}_{1}.$

  4. S-4.

The matrix $\mathbf{S}$ remains bounded near the points $z=-a,0$.

Before moving to the construction of the mentioned parametrices, we conclude
this section with the needed estimate for the jump matrix
$\mathbf{J}_{\mathbf{S}}$ away from $[-a,0]$. For that, recall the matrix norm
notation introduced in (6.1),(6.2),(6.3).

######  Proposition 10.1.

For $U_{0}$ as in Proposition 8.2, introduce the set

$\Gamma_{\varepsilon}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\Gamma_{\mathbf{S}}\setminus\left([-a,0]\cup
U_{0}\cup D_{\varepsilon}(-a)\right)$

For some $\varepsilon>0$, and possibly reducing $U_{0}$ if necessary, there is
an $\eta>0$ such that

$\|\mathbf{J}_{\mathbf{S}}-\mathbf{I}\|_{L^{1}\cap L^{2}\cap
L^{\infty}(\Gamma_{\varepsilon})}=\mathcal{O}(\operatorname{\rm e}^{-\eta
n}),$

as $n\to\infty$.

###### Proof.

From Proposition 8.1-(iv) we obtain that for any $\varepsilon>0$ there is a
constant $\eta^{\prime}>0$ for which

$\operatorname{Re}\phi_{+}(x)\geq\eta^{\prime},\quad\text{for every
}x\in(-a-\varepsilon,\varepsilon).$

On the other hand, the jump conditions on Proposition 8.1-(ii) combined with
Cauchy-Riemann equations imply in a standard way that
$\operatorname{Re}\phi\leq-\eta$ along the lipses of the lenses and away from
the endpoints $-a$ and $0$, as long as the lens stay within a positive but
small distance from the interval $[-a,0]$. From these pointwise estimates, the
growth of $\operatorname{Re}\phi$ as $z\to\pm\infty$ and the fact that
$\sigma_{n}$ remains bounded on $\mathbb{R}$ and $1/\sigma_{n}$ grows at most
with $\mathcal{O}(\operatorname{\rm e}^{cn^{2/3}})$ on compacts of
$\mathbb{C}$, the claimed $L^{p}$ estimates follow in a standard manner. ∎

###  10.3. Global parametrix

The global parametrix problem, obtained after neglecting the jumps of
$\mathbf{S}$ that are exponentially close to the identity, is the following
RHP.

  1. G-1.

$\mathbf{G}:\mathbb{C}\setminus[-a,0]\to\mathbb{C}^{2\times 2}$ is analytic.

  2. G-2.

For $z\in(-a,0)$, it satisfies the jump

$\mathbf{G}_{+}(z)=\mathbf{G}_{-}(z)\left(\sigma_{n}(z)\mathbf{E}_{12}-\frac{1}{\sigma_{n}(z)}\mathbf{E}_{21}\right).$

  3. G-3.

As $z\to\infty$,

$\mathbf{G}(z)=\mathbf{I}+\mathcal{O}(z^{-1}).$

  4. G-4.

$\mathbf{G}$ has square-integrable singularities at $z=-a,0$.

The construction of the global parametrix follows standard techniques. First,
one introduces a function that we denote $\mathsf{q}(z)$, with the aim at
transforming the RHP for $\mathbf{G}$ to a RHP with constant jumps. Then, by
diagonalizing the resulting jump matrix, we further reduce the problem to two
scalar-valued RHPs. With the help of Plemelj’s formula, we then solve these
scalar RHPs, and by tracing back all the transformations we recover the matrix
$\mathbf{G}$ itself.

The procedure just described is standard in RHP literature, see for instance
[7, Appendix A.1], so we refrain from completing it in detail and instead only
describe the final form of the solution.

The function $\sigma_{n}$ does not vanish and is real and positive over the
real axis, so its real logarithm over the real axis is well defined. With this
in mind, introduce

$\mathsf{q}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{((z+a)z)^{1/2}}{2\pi}\int_{-a}^{0}\frac{\log\sigma_{n}(x)}{\sqrt{|x|(x+a)}}\frac{\mathrm{d}x}{x-z},\quad
z\in\mathbb{C}\setminus[-a,0],$

where $(\cdot)^{1/2}$ stands for the principal branch of the square root and
$\sqrt{\cdot}$ is reserved for the standard positive real root of positive
real numbers. This function $\mathsf{q}$ depends on $n$ but we do not make
this dependence explicit for ease of notation. It is analytic on
$\mathbb{C}\setminus[-a,0]$, and it is chosen to satisfy the jump condition

$\mathsf{q}_{+}(x)+\mathsf{q}_{-}(x)=-\log\sigma_{n}(x),\quad-a<x<0.$

Furthermore, standard calculations show that

$\mathsf{q}(z)=\mathcal{O}(1),\quad
z\to-a,0,\qquad\text{and}\qquad\mathsf{q}(z)=\mathsf{q}_{0}+\frac{\mathsf{q}_{1}}{z}+\mathcal{O}(z^{-2}),\quad
z\to\infty,$

with coefficients given by

$\mathsf{q}_{0}=\mathsf{q}_{0}(n)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{1}{2\pi}\int_{-a}^{0}\frac{\log\sigma_{n}(x)}{\sqrt{|x|(x+a)}}\mathrm{d}x\quad\text{and}\quad\mathsf{q}_{1}=\mathsf{q}_{1}(n)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{1}{2\pi}\int_{-a}^{0}\frac{x\log\sigma_{n}(x)}{\sqrt{|x|(x+a)}}\mathrm{d}x+\frac{a\mathsf{q}_{0}}{2}.$
(10.3)

Next, set

$\mathbf{U}_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{pmatrix},\quad\mathsf{m}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{z}{z+a},$
(10.4)

which is consistent with (4.4), and introduce

$\mathbf{M}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{U}_{0}\mathsf{m}(z)^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}=\frac{1}{2}\begin{pmatrix}\mathsf{m}(z)+\dfrac{1}{\mathsf{m}(z)}&-\mathrm{i}\left(\mathsf{m}(z)-\dfrac{1}{\mathsf{m}(z)}\right)\\\
\mathrm{i}\left(\mathsf{m}(z)-\dfrac{1}{\mathsf{m}(z)}\right)&\mathsf{m}(z)+\dfrac{1}{\mathsf{m}(z)}\end{pmatrix}$
(10.5)

This matrix $\mathbf{M}$ satisfies

$\mathbf{M}_{+}(z)=\mathbf{M}_{-}(z)\left(\mathbf{E}_{12}-\mathbf{E}_{21}\right),\;-a<z<0,\quad\text{and}\quad\mathbf{M}(z)=\mathbf{I}-\frac{a}{4z}\bm{\sigma}_{2}+\mathcal{O}(z^{-2}),$

where $\bm{\sigma}_{2}$ is the second Pauli matrix (recall (3.2)).

Then the solution to the global parametrix RHP-$\mathbf{G}$ is

$\mathbf{G}(z)=\operatorname{\rm
e}^{-\mathsf{q}_{0}\bm{\sigma}_{3}}\mathbf{M}(z)\operatorname{\rm
e}^{\mathsf{q}(z)\bm{\sigma}_{3}},\quad z\in\mathbb{C}\setminus[-a,0].$ (10.6)

This solution $\mathbf{G}$ satisfies

$\mathbf{G}(z)=\mathsf{I}+\frac{\mathbf{G}_{1}}{z}+\mathcal{O}(z^{-2}),\;z\to\infty,\quad\text{with}\quad\mathbf{G}_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{pmatrix}\mathsf{q}_{1}&\dfrac{\mathrm{i}a}{4}\operatorname{\rm
e}^{-2\mathsf{q}_{0}}\\\ -\dfrac{\mathrm{i}a}{4}\operatorname{\rm
e}^{2\mathsf{q}_{0}}&-\mathsf{q}_{1}\end{pmatrix}.$ (10.7)

Recall that $U_{0}$ denotes the neighborhood of the origin given in
Proposition 8.2. We will also need some control on $\mathsf{q}$ inside
$U_{0}$.

For the next result, set

$\displaystyle
F_{\beta}(\mathsf{s})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{0}^{\infty}v^{\beta}\log(1+\operatorname{\rm
e}^{-\mathsf{s}-v})\mathrm{d}v,$ (10.8)
$\displaystyle\mathsf{q}_{0}^{(1)}=\mathsf{q}_{0}^{(1)}(\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\mathsf{t}^{1/2}}{2\pi
a^{1/2}}F_{-1/2}(\mathsf{s}),$
$\displaystyle\mathsf{q}^{(1)}(z)=\mathsf{q}^{(1)}(z\mid\mathsf{s},\mathsf{t})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\mathsf{t}^{1/2}}{2\pi
a^{1/2}\mathsf{m}(z)^{1/2}}F_{-1/2}(\mathsf{s}),$

which are $n$-independent quantities. The index $\beta$ does not have any
specific meaning for what comes later, but it arises naturally from the
asymptotic analysis resulting in the following result.

######  Lemma 10.2.

For any fixed $\mathsf{s}_{0}>0$, the estimate

$\mathsf{q}_{0}=\frac{1}{n^{1/3}}\mathsf{q}_{0}^{(1)}+\mathcal{O}(n^{-2/3}),\quad
n\to\infty,$

is valid uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$. In addition, the
estimates

$\mathsf{q}(z)=\frac{1}{n^{1/3}}\mathsf{q}^{(1)}(z)+\mathcal{O}(n^{-2/3}),\quad\text{and}\quad\mathsf{q}^{\prime}(z)=\mathcal{O}(n^{-1/3}),$

are valid uniformly for $z$ on compacts of $\mathbb{C}\setminus[-a,0]$ (in
particular on $\partial U_{0}$) and uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0}$, and carry through to boundary values
$\mathsf{q}_{\pm}(x)$ for $x$ along $\mathbb{R}\setminus\\{-a,0\\}$.

Finally,

$\mathbf{M}(z)=\left(\mathbf{I}+\frac{1}{n^{1/3}}\left(\mathsf{q}^{(1)}_{0}\bm{\sigma}_{3}-\mathsf{q}^{(1)}(z)\mathbf{M}(z)\bm{\sigma}_{3}\mathbf{M}(z)^{-1}\right)+\mathcal{O}(n^{-2/3})\right)\mathbf{G}(z),\quad
n\to\infty,$

uniformly for $z\in\partial U_{0}$ and $s\geq-\mathsf{s}_{0}$.

###### Proof.

The estimate for $\mathsf{q}_{0}(n)$ follows immediately from an application
of Proposition A.2. The estimates for $\mathsf{q}(z)$ and
$\mathsf{q}^{\prime}(z)$ also follow from Proposition A.2, once we observe
that the integrals defining them can be slightly deformed to the upper/lower
half plane in a neighborhood of the unique point in the intersection $\partial
U_{0}\cap(-a,0)$.

Finally, using the first part of the statement and the fact that $\mathbf{M}$
is bounded for $z\in\partial U_{0}$ and independent of $n$, we expand the
exponentials in series and write

$\displaystyle\mathbf{M}(z)\mathbf{G}(z)^{-1}$
$\displaystyle=\mathbf{M}(z)\operatorname{\rm
e}^{-\mathsf{q}(z)\bm{\sigma}_{3}}\mathbf{M}(z)^{-1}\operatorname{\rm
e}^{\mathsf{q}_{0}\bm{\sigma}_{3}}$
$\displaystyle=\left(\mathbf{I}-\mathsf{q}(z)\mathbf{M}(z)\bm{\sigma}_{3}\mathbf{M}(z)^{-1}+\mathcal{O}(n^{-2/3})\right)\left(\mathbf{I}+\mathsf{q}_{0}\bm{\sigma}_{3}+\mathcal{O}(n^{-2/3})\right),$

and the last claim follows after rearranging the terms in this expansion. ∎

###  10.4. Local Parametrix near $-a$

The local parametrix $\mathbf{P}=\mathbf{P}^{(a)}$ near $z=-a$ is constructed
in a neighborhood of $z=-a$ which without loss of generality can be taken to
be the disk $D_{\delta}(-a)$ of radius $\delta$ around $a$, and it is the
solution to the following RHP.

  1. $\mathbf{P}^{({a})}$-1.

The matrix
$\mathbf{P}^{(a)}:D_{\delta}(-a)\setminus\Gamma_{\mathbf{S}}\to\mathbb{C}^{2\times
2}$ is analytic.

  2. $\mathbf{P}^{({a})}$-2.

For $z\in\Gamma_{\mathbf{S}}\cap\partial D_{\delta}(-a)$, it satisfies the
jump
$\mathbf{P}^{(a)}_{+}(z)=\mathbf{P}^{(a)}_{-}(z)\mathbf{J}_{\mathbf{S}}(z)$.

  3. $\mathbf{P}^{({a})}$-3.

Uniformly for $z\in\partial D_{\delta}(-a)$,

$\mathbf{P}^{(a)}(z)=\left(\mathbf{I}+\mathit{o}(1)\right)\mathbf{G}(z),\quad
n\to\infty.$

  4. $\mathbf{P}^{({a})}$-4.

The matrix $\mathbf{P}^{(a)}$ remains bounded as $z\to-a$.

The asymptotic condition $\mathbf{P}^{({a})}$-3. above will be improved to
(10.13) below.

From the conditions on $Q$, we know that there exists a value $\eta>0$ for
which

$\operatorname{Re}Q(z)\geq 2\eta,\quad|z+a|<\delta.$

This value is uniform for $\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}],$
for any $\mathsf{t}_{0}\in(0,1)$ fixed, and it is independent of
$\mathsf{s}\in\mathbb{R}$. In particular, once we fix $\mathsf{s}_{0}>0$ and
assume that $\mathsf{s}\geq-\mathsf{s}_{0}$, from this inequality we obtain

$|\operatorname{\rm e}^{-\mathsf{s}-n^{2/3}Q(z)}|\leq\operatorname{\rm
e}^{-n^{2/3}\eta},\quad|z+a|<\delta,\quad\text{for large enough }n.$

This way, for $n>0$ sufficiently large the function $\sigma_{n}$ admits an
analytic continuation to the whole disk $D_{\delta}(-a)$, and this
continuation does not have zeros on the same disk. Thus, a branch of
$\log\sigma_{n}$ is well defined in a neighborhood of $z=-a$, and the just
mentioned estimate also shows that

$\log\sigma_{n}(z)=\mathcal{O}(\operatorname{\rm e}^{-\eta n^{2/3}}),\quad
n\to\infty,$ (10.9)

uniformly for $z$ in a neighborhood of $z=-a$ and
$\mathsf{s}\geq-\mathsf{s}_{0}$.

With this in mind, the parametrix $\mathbf{P}^{(a)}$ can be constructed
explicitly out of Airy functions in a standard way, see for instance [33,
Section 7.6]. Since it involves a somewhat nonstandard matching analytic
prefactor that accounts for $\sigma_{n}$, we briefly go over this
construction.

Recall the contour $\mathsf{\Sigma}$ introduced in (4.1). With appropriate
Airy functions, we construct a $2\times 2$ matrix
$\mathbf{\Psi}_{\operatorname{Ai}}$, which is analytic on
$\mathbb{C}\setminus\mathsf{\Sigma}$ and satisfies

$\mathbf{\Psi}_{\operatorname{Ai},+}(\zeta)=\mathbf{\Psi}_{\operatorname{Ai},-}(\zeta)\times\begin{cases}\mathbf{I}-\mathbf{E}_{12},&\zeta\in\mathsf{\Sigma}_{0},\\\
\mathbf{I}-\mathbf{E}_{21},&\zeta\in\mathsf{\Sigma}_{1}\cup\mathsf{\Sigma}_{3},\\\
-\mathbf{E}_{12}+\mathbf{E}_{21},&\zeta\in\mathsf{\Sigma}_{2},\\\ \end{cases}$

and

$\mathbf{\Psi}_{\operatorname{Ai}}(\zeta)=\zeta^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}\left(\mathbf{I}+\mathcal{O}(\zeta^{-3/2})\right)\operatorname{\rm
e}^{-\frac{2}{3}\zeta^{3/2}\bm{\sigma}_{3}},\quad\zeta\to\infty.$ (10.10)

In fact, $\mathbf{\Psi}_{\operatorname{Ai}}$ can be obtained with a
modification of the matrix ${\mathbf{\Phi}}_{\rm Ai}$ which we previously used
in (6.4). We will not need its explicit form, so we do not write it down
explicitly.

Using the properties of $\phi$ we construct a conformal map $\varphi$ from a
neighborhood of $-a$ to a neighborhood of the origin, with

$\varphi(-a)=0,\quad\varphi^{\prime}(-a)<0\quad\text{and}\quad\frac{2}{3}\varphi(z)^{3/2}=\phi(z)+2\pi\mathrm{i}\mathbb{Z},\;z\in
D_{\delta}(-a)\setminus\mathsf{\Sigma}_{\mathbf{S}}.$

With standard arguments (see for instance the proof of Proposition 10.5 below
for similar arguments), one shows that the matrix

$\mathbf{F}^{(a)}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{G}(z)\operatorname{\rm
e}^{\frac{1}{2}\log\sigma_{n}(z)\bm{\sigma}_{3}}\mathbf{U}_{0}^{-1}(n^{2/3}\varphi(z))^{-\bm{\sigma}_{3}/4}$
(10.11)

is analytic on a neighborhood of $z=-a$. The local parametrix then takes the
form

$\mathbf{P}^{(a)}(z)=\mathbf{F}^{(a)}(z)\mathbf{\Psi}_{\operatorname{Ai}}(n^{2/3}\varphi(z))\operatorname{\rm
e}^{-\frac{1}{2}\log\sigma_{n}(z)\bm{\sigma}_{3}}\operatorname{\rm
e}^{n\phi(z)\bm{\sigma}_{3}},\quad z\in
D_{\delta}(-a)\setminus\Gamma_{\mathbf{S}}.$ (10.12)

As a result, the error term in fact takes on the stronger form

$\mathbf{P}^{(a)}(z)=\mathbf{G}(z)\left(\mathbf{I}+\mathcal{O}(n^{-1})\right),\quad
n\to\infty,$ (10.13)

which is valid uniformly for $z\in\partial D_{\delta}(-a)$ and uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$, for any $\mathsf{s}_{0}>0$
and $\mathsf{t}_{0}\in(0,1)$ fixed.

###  10.5. Local Parametrix near the origin

The local parametrix near the origin requires the model problem from Section
4.

Recall the neighborhood $U_{0}$ of the origin introduced in Proposition 8.2.
The initial local parametrix we seek for should be the solution to the
following RHP.

  1. $\mathbf{P}^{(0)}$-1.

The matrix
$\mathbf{P}^{(0)}:U_{0}\setminus\Gamma_{\mathbf{S}}\to\mathbb{C}^{2\times 2}$
is analytic.

  2. $\mathbf{P}^{(0)}$-2.

For $z\in\Gamma_{\mathbf{S}}\cap U_{0}$, it satisfies the jump
$\mathbf{P}^{(0)}_{+}(z)=\mathbf{P}^{(0)}_{-}(z)\mathbf{J}_{\mathbf{S}}(z)$.

  3. $\mathbf{P}^{(0)}$-3.

Uniformly for $z\in\partial U_{0}$,

$\mathbf{P}^{(0)}(z)=\left(\mathbf{I}+\mathit{o}(1)\right)\mathbf{G}(z),\quad
n\to\infty.$

  4. $\mathbf{P}^{(0)}$-4.

$\mathbf{P}^{(0)}$ remains bounded as $z\to 0$.

To construct the solution $\mathbf{P}^{(0)}$ required above, some work is
needed. Aiming at removing $\phi$ from the jump of $\mathbf{P}^{(0)}$, we
change this RHP-$\mathbf{P}$ with the transformation

$\mathbf{L}(z)=\mathbf{P}^{(0)}(z)\operatorname{\rm
e}^{-n\phi(z)\bm{\sigma}_{3}},\quad z\in U_{0}\setminus\Gamma_{\mathbf{S}}.$
(10.14)

Then the matrix $\mathbf{L}$, should it exist, must satisfy the following RHP.

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

The matrix $\mathbf{L}:U_{0}\setminus\Gamma_{\mathbf{S}}\to\mathbb{C}^{2\times
2}$ is analytic.

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

For $z\in\Gamma_{\mathbf{S}}\cap U_{0}$, it satisfies the jump
$\mathbf{L}_{+}(z)=\mathbf{L}_{-}(z)\mathbf{J}_{\mathbf{L}}(z)$, with

$\mathbf{J}_{\mathbf{L}}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\sigma_{n}(z)\mathbf{E}_{12}-\frac{1}{\sigma_{n}(z)}\mathbf{E}_{21},&z\in
U_{0}\cap(-a,0),\\\
\mathbf{I}+\frac{1}{\sigma_{n}(z)}\mathbf{E}_{21},&z\in\partial\mathcal{G}^{\pm}\cap
U_{0},\\\ \mathbf{I}+\sigma_{n}(z)\mathbf{E}_{12},&z\in
U_{0}\cap(0,\infty).\end{dcases}$ (10.15)

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

Uniformly for $z\in\partial U_{0}$,

$\mathbf{L}(z)=\left(\mathbf{I}+\mathit{o}(1)\right)\mathbf{G}(z)\operatorname{\rm
e}^{-n\phi(z)\bm{\sigma}_{3}},\quad n\to\infty.$

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

The matrix $\mathbf{L}$ remains bounded as $z\to 0$.

Based on the usual way of matching the local parametrix with a model problem,
one is tempted to moving the non-constant part of the jump - namely
$\sigma_{n}$ \- to the behavior at $\partial U_{0}$ as well. This would be
done so including a term of the form
$\sigma_{n}^{\bm{\sigma}_{3}/2}=\operatorname{\rm
e}^{\bm{\sigma}_{3}\log\sigma_{n}/2}$ into the transformation
$\mathbf{P}\mapsto\mathbf{L}$, in much the same way we did in (10.12).
However, as we discussed in Section 3.1, for any $s\in\mathbb{R}$ fixed there
are poles of $\sigma_{n}$ accumulating too fast near the origin, so
$\sigma_{n}$ fails to be analytic in any small neighborhood of the origin and
we have to stick to the non-constant jumps as above.

The RHP-$\mathbf{L}$ has a solution if, and only if, RHP-$\mathbf{P}^{(0)}$
has a solution. Such solutions need not be unique, as one could possibly
improve on the asymptotic matching conditions on $\partial U_{0}$. The goal of
the rest of this section is to describe a solution $\mathbf{L}$, and
consequently a solution $\mathbf{P}^{(0)}$ related by (10.14), with a more
explicit control of the error term in RHP-$\mathbf{L}$-3. For that, we use the
model problem thoroughly studied in Sections 4 and 7.

The construction that follows needs several quantities that appeared before.
These are the conformal map $\psi$ appearing in Proposition 8.2, the function
$\mathsf{H}_{Q}$ introduced with the help of Proposition 8.3, the model RHP
solution $\mathbf{\Phi}=\mathbf{\Phi}(\cdot\mid\mathsf{h})$ introduced in
Section 4 and further discussed in Section 7, and the constant
$\mathsf{q}_{0}$ and matrices $\mathbf{U}_{0}$ and $\mathbf{M}(z)$ from
(10.3)–(10.5). With all these quantities at hand, we set

$\displaystyle\mathbf{L}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{F}_{n}(z)\mathbf{\Phi}_{n}(z),\quad
z\in U_{0}\setminus\mathsf{\Sigma}_{\mathbf{S}},\qquad\text{with the choices}$
(10.16)
$\displaystyle\widehat{\mathbf{\Phi}}_{n}(\zeta)=\mathbf{\Phi}\left(\zeta\mid\mathsf{h}=\mathsf{h}_{n}\right),\quad\mathsf{h}_{n}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathsf{s}+n^{2/3}\mathsf{H}_{Q}(\zeta/n^{2/3}),$
$\displaystyle{\mathbf{\Phi}}_{n}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\widehat{\mathbf{\Phi}}_{n}\left(\zeta=n^{2/3}\psi(z)\right),$
$\displaystyle{\mathbf{F}}_{n}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{M}(z)\mathbf{U}_{0}(n^{2/3}\psi(z))^{-\bm{\sigma}_{3}/4}=\mathbf{U}_{0}\mathsf{m}(z)^{\bm{\sigma}_{3}/4}(n^{2/3}\psi(z))^{-\bm{\sigma}_{3}/4}.$

With the identification $\tau=n^{2/3}$ and thanks to Proposition 8.3, the
function $\mathsf{h}_{n}$ becomes admissible in the sense of Definition 4.1,
so the notation for the corresponding solution
$\mathbf{\Phi}_{n}=\mathbf{\Phi}(\cdot\mid\mathsf{h}_{n})$ chosen above is
consistent with the solution
$\mathbf{\Phi}_{\tau}(\zeta)\big{|}_{\tau=n^{2/3}}$ in (4.5). For later
reference, we keep track of the expansion

$\mathsf{h}_{n}(\zeta)=\mathsf{s}-\mathsf{c}_{\mathsf{H}}\zeta+\mathcal{O}(\zeta^{-2}),\quad\zeta\to
0,\quad\text{where we recall
that}\quad\mathsf{c}_{\mathsf{H}}=\frac{\mathsf{t}}{\mathsf{c}_{V}},$ (10.17)

compare (8.7) with Definition 4.1–(ii).

######  Proposition 10.3.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$. There exists
$n_{0}=n_{0}(\mathsf{s}_{0},\mathsf{t}_{0})$ for which for any
$s\geq-\mathsf{s}_{0}$ and any
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$ the matrix
${\mathbf{\Phi}}_{n}$ exists for every $n\geq n_{0}$. This matrix satisfies
the jump

${\mathbf{\Phi}}_{n,+}(z)={\mathbf{\Phi}}_{n,-}(z)\mathbf{J}_{\mathbf{L}}(z),\quad
z\in U_{0}\cap\Gamma_{\mathbf{S}}.$

Furthermore, for the matrix

${\mathbf{\Phi}}^{(1)}_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{\Phi}^{(1)}(\mathsf{h}=\mathsf{h}_{n}),\quad\text{with
}\mathbf{\Phi}^{(1)}(\mathsf{h})\text{ as in \eqref{eq:ModelRHPAsymp}},$

the asymptotic expansion

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

holds true uniformly for $z\in\partial U_{0}$ and uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

###### Proof.

The existence of ${\mathbf{\Phi}}_{n}$ is granted by the first claim of
Theorem 7.1. The jumps for ${\mathbf{\Phi}}_{n,+}$ follow from the jumps in
(4.2), the definition of $\mathsf{h}_{n}$ taken in (10.16), the correspondence
(8.6) and the conformality of the change of variables $\zeta=n^{2/3}\psi(z)$.
The asymptotic expansion for ${\mathbf{\Phi}}_{n}$ is immediate from (4.3) ∎

The introduction of the additional notation $\widehat{\mathbf{\Phi}}_{n}$,
which plays the role of the local parametrix in the variable $\zeta$, is
convenient for later calculations. At that moment, some of its properties will
be needed, and we keep track of these properties with the next result. For the
formal statement, we recall that $\Phi(\xi\mid\mathsf{S},\mathsf{T})$ is the
solution to the integro-differential PII that already appeared in (2.11) and
(5.15), and $\mathbf{\Phi}_{0}^{(1)}$ is the residue matrix from (5.17) that
collects the functions $\mathsf{P}(\mathsf{S},\mathsf{T})$ and
$\mathsf{Q}(\mathsf{S},\mathsf{T})$ which, in turn, are related to $\Phi$ as
explained in (5.13) et seq..

######  Proposition 10.4.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$ and $\nu\in(0,2/3)$, and
let $\mathsf{c}_{\mathsf{H}}$ be as in (8.7). The following asymptotic
formulas hold true uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$.

The matrix $\mathbf{\Phi}_{n}^{(1)}$ from Proposition 10.3 satisfies

$\left({\mathbf{\Phi}}_{n}^{(1)}\right)_{21}=\frac{\mathrm{i}}{\mathsf{c}_{\mathsf{H}}^{1/2}}\left(\mathsf{p}(\mathsf{s},\mathsf{c}_{\mathsf{H}})-\frac{\mathsf{s}^{2}}{4\mathsf{c}_{\mathsf{H}}^{3/2}}\right)+\mathcal{O}(n^{-\nu}),\quad
n\to\infty.$ (10.18)

Furthermore, for ${\mathbf{\Phi}}_{0}$ being the solution to the RHP in
(5.9)–(5.10), the estimate

$\widehat{\mathbf{\Phi}}_{n,+}(\zeta)=(\mathbf{I}+\mathcal{O}(n^{-\nu})){\mathbf{\Phi}}_{0,+}(\zeta\mid\mathsf{s},\mathsf{c}_{\mathsf{H}}),\quad
n\to\infty,$ (10.19)

holds true uniformly for $\zeta\in\mathsf{\Sigma}$ with $|\zeta|\leq
n^{\frac{1}{3}-\frac{1}{2}\nu}$.

###### Proof.

It is immediate from the identification $\tau=n^{2/3}$, Equations (10.16) and
Theorem 7.1. ∎

Next, we now verify that $\mathbf{L}$ given as in (10.16) indeed solves the
RHP-$\mathbf{L}$.

######  Proposition 10.5.

The matrix $\mathbf{L}$ solves the RHP-$\mathbf{L}$.

Furthermore, setting

$\mathbf{L}_{1}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\left({\mathbf{\Phi}}^{(1)}_{n}\right)_{21}}{\psi(z)^{1/2}}\mathbf{M}(z)\mathbf{U}_{0}\mathbf{E}_{21}\mathbf{U}_{0}^{-1}\mathbf{M}(z)^{-1},\quad
z\in\partial U_{0},$ (10.20)

the condition $\mathbf{L}$-3. is improved to

$\mathbf{L}(z)=\left(\mathbf{I}+\frac{1}{n^{1/3}}\mathbf{L}_{1}(z)+\mathcal{O}(n^{-2/3})\right)\mathbf{M}(z)\operatorname{\rm
e}^{-n\phi(z)\bm{\sigma}_{3}},\quad n\to\infty,$ (10.21)

uniformly for $z\in\partial U_{0}$ and $\mathsf{s}\geq-\mathsf{s}_{0}$,
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$, for any
$\mathsf{s}_{0}>0,\mathsf{t}_{0}\in(0,1)$.

###### Proof.

First we prove that ${\mathbf{F}}$ is analytic. For that, notice that a jump
for it may come only from the factors $\mathbf{M}$ and $\psi^{1/4}$, and
therefore only possibly in the interval $(-a,0)\cap U_{0}$. However, along
this interval it is simple to compute that
$(\psi^{1/4})_{\pm}=\operatorname{\rm e}^{\pm\pi\mathrm{i}/4}|\psi|^{1/4}$,
and also that

$\mathbf{U}_{0}^{-1}\mathbf{J}_{\mathbf{M}}(x)\mathbf{U}_{0}=\mathrm{i}\bm{\sigma}_{3},$

from which we obtain

${\mathbf{F}}_{-}(x)^{-1}{\mathbf{F}}_{+}(x)=n^{\bm{\sigma}_{3}/6}|\psi|^{\bm{\sigma}_{3}/4}\operatorname{\rm
e}^{-\pi\mathrm{i}\bm{\sigma}_{3}/4}\mathrm{i}\bm{\sigma}_{3}\operatorname{\rm
e}^{-\pi\mathrm{i}\bm{\sigma}_{3}/4}|\psi|^{-\bm{\sigma}_{3}/4}n^{-\bm{\sigma}_{3}/6}=\mathbf{I},$

so ${\mathbf{F}}$ is indeed analytic across $(-a,0)\cap U_{0}$. In principle,
${\mathbf{F}}$ may have an isolated singularity at $0$, but because

$\mathbf{M}(z),\mathbf{\varphi}(z)^{1/4}=\mathcal{O}(z^{1/4})\quad\text{as
}z\to 0,$

we see that $z=0$ is a removable singularity. In virtue of the definition of
$\mathbf{L}$ in (10.16) and the jump for $\mathbf{\Phi}_{n}$ from Proposition
10.3, the analyticity of $\mathbf{F}$ is enough to conclude that $\mathbf{L}$
satisfies RHP.$\mathbf{L}$-1.

Knowing that ${\mathbf{F}}$ is analytic, the jump for $\mathbf{L}$ is
precisely the same as the jump for ${\mathbf{\Phi}}_{n}$, and by Proposition
10.3 we thus have that $\mathbf{L}$ satisfies RHP.$\mathbf{L}$-2.

Finally, to the asymptotic condition (10.20). For that, we use the asymptotic
condition for ${\mathbf{\Phi}}_{n}$ given by Proposition 10.3 and the
definition of ${\mathbf{F}}$ and write

$\mathbf{L}(z)=\mathbf{M}(z)\mathbf{U}_{0}n^{-\bm{\sigma}_{3}/6}\psi(z)^{-\bm{\sigma}_{3}/4}\left(\mathbf{I}+\frac{1}{n^{2/3}\psi(z)}{\mathbf{\Phi}}^{(1)}_{n}+\mathcal{O}(n^{-4/3})\right)\\\
\times
n^{\bm{\sigma}_{3}/6}\psi(z)^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\operatorname{\rm
e}^{-n\phi(z)\bm{\sigma}_{3}},\quad n\to\infty,$

and where the error is uniform for $z\in\partial U_{0}$ and
$\mathsf{s},\mathsf{t}$ as claimed. Since $\partial U_{0}$ remains within a
positive distance from the unique zero $z=0$ of $\psi$, the function
$|\psi^{1/4}|$ remains bounded from below away from zero, so the corresponding
conjugation of the error by the term $\psi^{\bm{\sigma}_{3}/4}$ does not
change the order of the error. Next, the conjugation by
$n^{\bm{\sigma}_{3}/6}$ contributes at most to an error of order $n^{1/3}$,
and only in the $(2,1)$-entry. The remaining term $\mathbf{M}$ is bounded
along $\partial U_{0}$, so it can be commuted with the error term above
without changing its order, leading to (10.21). And (10.21) is indeed an
improvement of the asymptotic condition RHP.$\mathbf{L}$-3, because from Lemma
10.2 we know that $\mathbf{M}=(\mathbf{I}+\mathcal{O}(n^{-1/3}))\mathbf{G}$
uniformly as claimed. ∎

We now trace back the transformation $\mathbf{L}\mapsto\mathbf{P}^{(0)}$ and
construct the latter as

$\mathbf{P}^{(0)}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}={\mathbf{F}}_{n}(z){\mathbf{\Phi}}_{n}(z)\operatorname{\rm
e}^{n\phi(z)\bm{\sigma}_{3}},\quad z\in
U_{0}\setminus\mathsf{\Sigma}_{\mathbf{S}},$ (10.22)

keeping track that ${\mathbf{\Phi}}_{n}$ and $\mathbf{F}_{n}$ are introduced
in (10.16). With this construction and thanks to Lemma 10.2 and (10.21), the
matching condition RHP.$\mathbf{P}^{(0)}$-3 is

$\mathbf{P}^{(0)}(z)=\left(\mathbf{I}+\frac{1}{n^{1/3}}\left(\mathbf{L}_{1}(z)+\mathsf{q}^{(1)}_{0}\bm{\sigma}_{3}-\mathsf{q}^{(1)}(z)\mathbf{M}(z)\bm{\sigma}_{3}\mathbf{M}(z)^{-1}\right)+\mathcal{O}(n^{-2/3})\right)\mathbf{G}(z),\quad
n\to\infty,$ (10.23)

and the errors are uniform for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$.

###  10.6. Final transformation

The final transformation combines the local and global parametrices to remove
the non-decaying jumps from $\mathbf{S}$.

Set

$U\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=U_{0}\cup
D_{\delta}(-a),$

orienting $\partial U_{0}$ and $\partial D_{\delta}(-a)$ in the clockwise
direction. With $\mathbf{P}^{(a)}$ being the local parametrix near $-a$,
$\mathbf{P}^{(0)}$ the local parametrix near the origin and $\mathbf{G}$ the
global parametrix, we introduce a last parametrix $\mathbf{P}$ with unified
notation as

$\mathbf{P}(z)=\begin{cases}\mathbf{P}^{(a)}(z),&z\in
D_{\delta}(-a)\setminus\Gamma_{\mathbf{S}},\\\ \mathbf{P}^{(0)}(z),&z\in
U_{0}\setminus\Gamma_{\mathbf{S}},\\\ \mathbf{G}(z),&\text{elsewhere on
}\mathbb{C}\setminus(\Gamma_{\mathbf{S}}\cup\partial U).\end{cases}$

The final transformation $\mathbf{S}\mapsto\mathbf{R}$ is then

$\mathbf{R}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{S}(z)\mathbf{P}(z)^{-1},\quad
z\in\mathbb{C}\setminus(\Gamma_{\mathbf{S}}\cup\partial U).$

With this transformation, the jumps that $\mathbf{S}$ has in common with the
parametrices $\mathbf{G}$ and $\mathbf{P}$ get canceled, and we remain with
jumps only away from $[-a,0]$ and $U$. With

$\Gamma_{\mathbf{R}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\partial
U\cup\Gamma_{\mathbf{S}}\setminus\left([-a,0]\cup U\right),$

the matrix $\mathbf{R}$ satisfies the following RHP.

  1. R-1.

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

  2. R-2.

For $z\in\Gamma_{\mathbf{R}}$, it satisfies the jump
$\mathbf{R}_{+}(z)=\mathbf{R}_{-}(z)\mathbf{J}_{\mathbf{R}}(z)$, with

$\mathbf{J}_{\mathbf{R}}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{dcases}\mathbf{G}(z)\mathbf{J}_{\mathbf{S}}(z)\mathbf{G}(z)^{-1},&z\in\Gamma_{\mathbf{R}}\setminus\partial
U,\\\ \mathbf{P}^{(0)}(z)\mathbf{G}(z)^{-1},&z\in\partial U_{0},\\\
\mathbf{P}^{(a)}(z)\mathbf{G}(z)^{-1},&z\in\partial
D_{\delta}(-a),\end{dcases}$ (10.24)

  3. R-3.

With
$\mathbf{R}_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{S}_{1}-\mathbf{G}_{1}$,

$\mathbf{R}(z)=\mathbf{I}+\frac{1}{z}\mathbf{R}_{1}+\mathcal{O}(1/z^{2}),\quad
z\to\infty.$ (10.25)

To conclude that $\mathbf{R}$ is asymptotically close to the identity, we
control its jumps. We use the matrix norm notation introduced in (6.1)–(6.3).

######  Proposition 10.6.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$. There is $\eta>0$ for
which the jump matrix for $\mathbf{R}$ satisfies the estimates

$\|\mathbf{J}_{\mathbf{R}}-\mathbf{I}\|_{L^{1}\cap L^{2}\cap
L^{\infty}(\Gamma_{\mathbf{R}}\setminus\partial
U)}=\mathcal{O}(\operatorname{\rm e}^{-\eta n})$

and

$\|\mathbf{J}_{\mathbf{R}}-\mathbf{I}\|_{L^{1}\cap L^{2}\cap
L^{\infty}(\partial
U_{0})}=\mathcal{O}(n^{-1/3}),\quad\|\mathbf{J}_{\mathbf{R}}-\mathbf{I}\|_{L^{1}\cap
L^{2}\cap L^{\infty}(\partial D_{\delta}(-a))}=\mathcal{O}(n^{-1})$

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

###### Proof.

The function $\mathbf{G}$ remains uniformly bounded away from the interval
$[-a,0]$, and the first claimed decay then follows from (10.24) and
Proposition 10.1.

Using (10.13) and (10.24), we obtain the uniform estimate of the jump along
$\partial D_{\delta}(-a)$ directly. Since $\partial D_{\delta}(-a)$ is
bounded, the $L^{1}$ and $L^{2}$ estimates also follow. Finally, estimate for
the jump along $\partial U_{0}$ follows similarly, once we recall (10.23). ∎

With Proposition 10.6 at hand, the small norm theory of Riemann-Hilbert
problems yields

######  Theorem 10.7.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}>0$. The matrix $\mathbf{R}$
satisfies

$\mathbf{R}(z)=\mathbf{I}+\mathcal{O}(n^{-1/3}),\quad\mathbf{R}^{\prime}(z)=\mathcal{O}(n^{-1/3})\quad\text{and}\quad\mathbf{R}(w)^{-1}\mathbf{R}(z)=\mathbf{I}+\mathcal{O}\left(\frac{z-w}{n^{1/3}}\right),\quad
n\to\infty,$

and the residue matrix $\mathbf{R}_{1}$ satisfies

$\mathbf{R}_{1}=-\frac{1}{2\pi\mathrm{i}}\int_{\partial
U_{0}}\mathbf{R}_{-}(s)(\mathbf{J}_{\mathbf{R}}(s)-\mathbf{I})\mathrm{d}s+\mathcal{O}(n^{-1}),$
(10.26)

where the error terms are all uniform for $z,w$ on the same connected
components of $\mathbb{C}\setminus\Gamma_{\mathbf{R}}$, and they are also
uniform for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$.

###### Proof.

The arguments are standard, so we only sketch them. The small norm theory for
RHPs ensures the representation

$\mathbf{R}(z)=\mathbf{I}+\frac{1}{2\pi\mathrm{i}}\int_{\Gamma_{\mathbf{R}}}\frac{\mathbf{R}_{-}(s)(\mathbf{J}_{\mathbf{R}}(s)-\mathbf{I})}{s-z}\mathrm{d}s,\quad
z\in\mathbb{C}\setminus\Gamma_{\mathbf{R}}.$ (10.27)

This ensures the estimates for $\mathbf{R}$ and $\mathbf{R}^{\prime}$ away
from $\Gamma_{\mathbf{R}}$ and, because the jump matrix is analytic on a
neighborhood of $\Gamma_{\mathbf{R}}$, we are able to extend these estimates
also to $\Gamma_{\mathbf{R}}$. To obtain the estimate involving $z$ and $w$,
we then write, with the help of Cauchy’s formula,

$\mathbf{R}(w)^{-1}\mathbf{R}(z)=\mathbf{I}+\mathbf{R}(w)^{-1}\left(\mathbf{R}(w)-\mathbf{R}(z)\right)=\mathbf{I}+\mathbf{R}(w)^{-1}\frac{w-z}{2\pi\mathrm{i}}\oint\frac{\mathbf{R}(s)}{(s-w)(s-z)}\mathrm{d}s,$
(10.28)

where the integral is over a contour on
$\mathbf{C}\setminus\Gamma_{\mathbf{R}}$ encircling both $w$ and $z$. The
right-hand side is now $\mathbf{I}+\mathcal{O}((w-z)n^{-1/3})$ by the estimate
on $\mathbf{R}$ already proven.

Finally, the estimate for $\mathbf{R}_{1}$ follows expanding (10.27) as
$z\to\infty$, and then using that the jump matrix decays to the identity at
least as $\mathcal{O}(n^{-1})$ away from $\partial U_{0}$. ∎

Later on, we also need the first term in the asymptotic expansion for
$\mathbf{R}$, we state it as a separate result.

######  Proposition 10.8.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}>0$. Setting

$\widehat{\mathbf{R}}_{1}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{2\pi\mathrm{i}}\int_{\partial
U_{0}}\left(\frac{(\widehat{\mathbf{\Phi}}_{n}^{(1)})_{21}}{\psi(s)^{1/2}\mathsf{m}(s)^{1/2}}\mathbf{E}_{21}-\mathsf{q}^{(1)}(s)\bm{\sigma}_{3}\right)\frac{\mathrm{d}s}{s-z}+\mathsf{q}_{0}^{(1)}\bm{\sigma}_{2},\quad
z\in U_{0},$ (10.29)

the matrix $\mathbf{R}$ has the expansion

$\mathbf{R}(z)=\mathbf{I}+\frac{1}{n^{1/3}}\mathbf{U}_{0}\widehat{\mathbf{R}}_{1}(z)\mathbf{U}_{0}^{-1}+\mathcal{O}(n^{-2/3}),\quad
n\to\infty$ (10.30)

where the error term is uniform for $z\in U_{0}$, and it is also uniform for
$\mathsf{s}\geq-\mathsf{s}_{0}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$.

###### Proof.

Starting from the representation (10.27), we write

$\mathbf{R}(z)-\mathbf{I}=\frac{1}{2\pi\mathrm{i}}\int_{\partial
U_{0}}\frac{\mathbf{R}_{-}(s)(\mathbf{J}_{\mathbf{R}}(s)-\mathbf{I})}{s-z}\mathrm{d}s+\mathcal{O}(n^{-1})=\frac{1}{2\pi\mathrm{i}}\int_{\partial
U_{0}}\frac{\mathbf{J}_{\mathbf{R}}(s)-\mathbf{I}}{s-z}\mathrm{d}s+\mathcal{O}(n^{-2/3}),$

and the result follows combining (10.24) and (10.23) and performing
straightforward calculations. ∎

All the transformations
$\mathbf{Y}\mapsto\mathbf{T}\mapsto\mathbf{S}\mapsto\mathbf{R}$ involve only
analytic factors in their construction. As such, we can actually slightly
deform the contour $\Gamma_{\mathbf{R}}$ in all these steps, so that in fact
the estimates in the previous result are valid everywhere on $\mathbb{C}$,
interpreting them as with boundary values when $z,w\in\Gamma_{\mathbf{R}}$. We
will use this fact without further warning.

With this theorem at hand, in the next sections we recover the needed
asymptotic formulas for the proof of our main results.

##  11\. Proof of main results

The next step in the direction of concluding our asymptotic analysis is to
unravel the transformations
$\mathbf{R}\mapsto\mathbf{S}\mapsto\mathbf{T}\mapsto\mathbf{Y}$. Introduce

$\mathbf{\Lambda}_{n}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{I}+\frac{\operatorname{\rm
e}^{2n\phi_{+}(x)}}{\sigma_{n}(x)}\chi_{(-a,0)}(x)\mathbf{E}_{21}\quad\text{and}\quad\mathbf{\Delta}_{n}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(\mathbf{I}+\frac{\chi_{(-a,0)}(x)}{\sigma_{n}(x)}\mathbf{E}_{21}\right),\quad
x\in\mathbb{R},$

which are related by

$\mathbf{\Lambda}_{n}(x)\operatorname{\rm
e}^{-n\phi_{+}(x)\bm{\sigma}_{3}}=\operatorname{\rm
e}^{-n\phi_{+}(x)\bm{\sigma}_{3}}\mathbf{\Delta}_{n}(x).$

Then the result of the unfolding of the transformations is

$\mathbf{Y}_{+}(z)=\operatorname{\rm
e}^{n\ell_{V}\bm{\sigma}_{3}}\mathbf{R}_{+}(x)\mathbf{P}_{+}(x)\mathbf{\Lambda}_{n}(x)\operatorname{\rm
e}^{-n(\phi_{+}(x)-V(x)/2)\bm{\sigma}_{3}},\quad x\in\mathbb{R}.$ (11.1)

We split the proofs of our results in the next few sections.

###  11.1. Proof of Theorem 2.4

Thanks to (11.1) and Theorem 10.7, the expression (9.10) reduces to the
asymptotic formula

$\operatorname{\rm
e}^{-n(V(x)+V(y))/2}\mathsf{K}_{n}^{Q}(x,y)=\frac{\operatorname{\rm
e}^{-n(\phi_{+}(x)+\phi_{+}(y))}}{2\pi\mathrm{i}(x-y)}\mathbf{e}^{\mathrm{T}}_{2}\mathbf{\Lambda}_{n}(y)^{-1}\mathbf{P}_{+}(y)^{-1}\left(\mathbf{I}+\mathcal{O}\left(\frac{x-y}{n^{1/3}}\right)\right)\mathbf{P}_{+}(x)\mathbf{\Lambda}_{n}(x)\mathbf{e}_{1},$
(11.2)

valid as $n\to\infty$, $x,y\in\mathbb{R}$ and uniformly for
$\mathsf{s},\mathsf{t}$ as in Theorem 10.7.

For the value $\mathsf{c}_{V}>0$ introduced in (8.5), we scale $x=u_{n}$ and
$y=v_{n}$ as in (2.19), where $u,v$ are on any given compact of the real axis.
For such values of $u$, for $n$ large enough the points $u_{n}$ are always on
the neighborhood $U_{0}$ of the origin where $\mathbf{P}=\mathbf{P}^{(0)}$,
and from (10.22) we simplify

$\mathbf{e}_{2}^{\mathrm{T}}\mathbf{\Lambda}_{n}^{-1}\mathbf{P}_{+}^{-1}=\operatorname{\rm
e}^{n\phi_{+}}\mathbf{e}_{2}^{\mathrm{T}}\mathbf{\Delta}_{n}^{-1}\mathbf{\Phi}_{n,+}^{-1}\mathbf{F}_{n}^{-1}\quad\text{and}\quad\mathbf{P}_{+}\mathbf{\Lambda}_{n}\mathbf{e}_{1}=\mathbf{F}_{n}\mathbf{\Phi}_{n,+}\mathbf{\Delta}_{n}\mathbf{e}_{1}\operatorname{\rm
e}^{n\phi_{+}}.$

where all the quantities above are evaluated at $u_{n}$. The next step is to
plug these identities into (11.2), when doing so we also use the estimates

$\mathbf{F}_{n}(u_{n})=\mathbf{U}_{0}\left(\mathbf{I}+\mathcal{O}(n^{-2/3})\right)(\mathsf{c}_{V}a)^{-\bm{\sigma}_{3}/4}n^{-\bm{\sigma}_{3}/6},\quad
n\to\infty,$

which follows from (10.16), (10.4) and (8.5), and its immediate consequence

$\mathbf{F}_{n}(v_{n})^{-1}\mathbf{F}_{n}(u_{n})=n^{\bm{\sigma}_{3}/6}\left(\mathbf{I}+\mathcal{O}\left(\frac{u-v}{n^{2/3}}\right)\right)n^{-\bm{\sigma}_{3}/6}=\mathbf{I}+\mathcal{O}\left(\frac{u-v}{n^{1/3}}\right),\quad
n\to\infty,$

which is obtained with arguments similar to (10.28). These estimates are
uniform for $u,v$ in compacts of $\mathbb{R}$, and are also uniform for
$\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}\in[\mathsf{t}_{0},1/\mathsf{t}_{0}]$. Equation (11.2) then
simplifies to

$\operatorname{\rm
e}^{-n(V(u_{n})+V(v_{n}))/2}\mathsf{K}_{n}^{Q}(u_{n},v_{n})=\\\
\frac{\mathsf{c}_{V}n^{2/3}}{2\pi\mathrm{i}(u-v)}\mathbf{e}_{2}^{\mathrm{T}}\mathbf{\Delta}_{n}(v_{n})^{-1}\mathbf{\Phi}_{n,+}(v_{n})^{-1}\left(\mathbf{I}+\mathcal{O}\left(\frac{u-v}{n^{1/3}}\right)\right)\mathbf{\Phi}_{n,+}(u_{n})\mathbf{\Delta}_{n}(u_{n})\mathbf{e}_{1}.$

Now, with the definition of $\mathbf{\Phi}_{n}$ in (10.16) at hand, the aid of
(10.19) and the constant $\mathsf{c}_{\mathsf{H}}$ appearing in (10.17), we
get

$\mathbf{\Phi}_{n,+}(u_{n})=\widehat{\mathbf{\Phi}}_{n,+}(n^{2/3}\psi(u_{n}))=\left(\mathbf{I}+\mathcal{O}(n^{-\nu})\right){\mathbf{\Phi}}_{0,+}(u\mid\mathsf{s},\mathsf{c}_{\mathsf{H}}),\quad
n\to\infty,$

for any $\nu\in(0,2/3)$, with the error being valid uniformly for $u$ in
compacts of $\mathbb{R}$, and also uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0},\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$. Also, thanks to Theorem 5.1 we know that
${\mathbf{\Phi}}_{0,+}(u\mid\mathsf{s},\mathsf{c}_{\mathsf{H}})^{\pm 1}$
remain bounded for $u$ in compacts and
$\mathsf{s}\geq-\mathsf{s}_{0},\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$. Therefore,

$\mathbf{\Phi}_{n,+}(v_{n})^{-1}\left(\mathbf{I}+\mathcal{O}\left(\frac{u-v}{n^{1/3}}\right)\right)\mathbf{\Phi}_{n,+}(u_{n})=\left(\mathbf{I}+\mathcal{O}(n^{-\nu})\right){\mathbf{\Phi}}_{0,+}(v\mid\mathsf{s},\mathsf{c}_{\mathsf{H}})^{-1}{\mathbf{\Phi}}_{0,+}(u\mid\mathsf{s},\mathsf{c}_{\mathsf{H}})+\mathcal{O}\left(\frac{u-v}{n^{1/3}}\right)$

as $n\to\infty$. In addition, the estimate

$\frac{1}{\sigma_{n}(u_{n})}=1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{t}u/\mathsf{c}_{V}}(1+\mathcal{O}(n^{-2/3}))=1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{c}_{\mathsf{H}}u}(1+\mathcal{O}(n^{-2/3})),\quad
n\to\infty,$ (11.3)

is valid uniformly for $u$ in compacts, uniformly for
$\mathsf{s}\in\mathbb{R}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$, and is immediate from (2.17). Combining everything, and
denoting $\chi_{+}=\chi_{[0,\infty)}$, we obtained the asymptotic formula

$\frac{\operatorname{\rm
e}^{-n(V(u_{n})+V(v_{n}))/2}}{\mathsf{c}_{V}n^{2/3}}\mathsf{K}_{n}^{Q}(u_{n},v_{n})=\frac{1+\mathcal{O}(n^{-\nu})}{2\pi\mathrm{i}(u-v)}\mathbf{e}_{2}^{\mathrm{T}}\left(\mathbf{I}-(1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{c}_{\mathsf{H}}v})\chi_{+}(v)\mathbf{E}_{21}\right)\\\
\times{\mathbf{\Phi}}_{0,+}(v\mid\mathsf{s},\mathsf{c}_{\mathsf{H}})^{-1}{\mathbf{\Phi}}_{0,+}(u\mid\mathsf{s},\mathsf{c}_{\mathsf{H}})\left(\mathbf{I}+(1+\operatorname{\rm
e}^{-\mathsf{s}+\mathsf{c}_{\mathsf{H}}u})\chi_{+}(u)\mathbf{E}_{21}\right)\mathbf{e}_{1}+\mathcal{O}(n^{-1/3}),\quad
n\to\infty,$

which is valid uniformly for $u,v$ in compacts of $\mathbb{R}$, and also
uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$. With
${\mathbf{\Delta}}_{0}$ as in (5.7), this identity rewrites as

$\frac{\operatorname{\rm
e}^{-n(V(u_{n})+V(v_{n}))/2}}{\mathsf{c}_{V}n^{2/3}}\mathsf{K}_{n}^{Q}(u_{n},v_{n})=\frac{1+\mathcal{O}(n^{-\nu})}{2\pi\mathrm{i}(u-v)}\\\
\times\left[\left({\mathbf{\Phi}}_{0}(v\mid\mathsf{s},\mathsf{c}_{\mathsf{H}}){\mathbf{\Delta}}_{0}(v\mid\mathsf{s},\mathsf{c}_{\mathsf{H}})\right)^{-1}{\mathbf{\Phi}}_{0}(u\mid\mathsf{s},\mathsf{c}_{\mathsf{H}}){\mathbf{\Delta}}_{0}(u\mid\mathsf{s},\mathsf{c}_{\mathsf{H}})\right]_{21,+}+\mathcal{O}(n^{-1/3}),\quad
n\to\infty,$

and the proof of Theorem 2.4 is now completed using Equation (5.16).

###  11.2. Proof of Theorem 2.5

Now that the RHP asymptotic analysis is completed, the proof of Theorem 2.5
follows standard steps. In our case, there is an additional cancellation that
has to be accounted for at a later step, in virtue of the presence of the
factor $\mathsf{q}$ in both the global and local parametrices, see (10.6) and
(10.23). So we opt for presenting the detailed calculation.

Starting from the representation (9.8), we unravel the transformations and
obtain

$-2\pi\mathrm{i}\left(\upgamma_{n-1}^{(n,Q)}\right)^{2}=\operatorname{\rm
e}^{-2n\ell_{V}}\left(\mathbf{R}_{1}+\mathbf{G}_{1}\right)_{21}=\operatorname{\rm
e}^{-2n\ell_{V}}\left((\mathbf{R}_{1})_{21}-\frac{\mathrm{i}a}{4}-\frac{1}{n^{1/3}}\frac{\mathsf{t}^{1/2}a^{1/2}}{4\pi\mathrm{i}}F_{-1/2}(\mathsf{s})+\mathcal{O}(n^{-2/3})\right),$
(11.4)

with $\mathbf{R}_{1}$ as in (10.25), and where for the second identity we used
the definition of $\mathbf{G}_{1}$ in (10.7) and the estimate for
$\mathsf{q}_{0}$ from Lemma 10.2.

It remains to estimate $\mathbf{R}_{1}$, which we do so starting from (10.26).
Using Cauchy-Schwartz, Propositions 10.6 and Theorem 10.7, we write

$\mathbf{R}_{1}=-\frac{1}{2\pi\mathrm{i}}\int_{\partial
U_{0}}(\mathbf{J}_{\mathbf{R}}(s)-\mathbf{I})\mathrm{d}s+\mathcal{O}(n^{-2/3}),\quad
n\to\infty.$

The matrix $\mathbf{J}_{\mathbf{R}}$ is in (10.24), and combining its explicit
expression along $\partial U_{0}$ with (10.23) and (10.5), after a cumbersome
but straightforward calculation we arrive at

$\left(\mathbf{R}_{1}\right)_{21}=-\frac{1}{2\pi\mathrm{i}n^{1/3}}\int_{\partial
U_{0}}\left(\frac{\mathrm{i}\mathsf{q}^{(1)}(s)(\mathsf{m}(s)-1)}{2\mathsf{m}(s)^{1/2}}+(\mathbf{L}_{1}(s))_{21}\right)\mathrm{d}s+\mathcal{O}(n^{-2/3}),\quad
n\to\infty,$

where we recall that $\mathsf{m}$, $\mathsf{q}^{(1)}$ and $\mathbf{L}_{1}$ are
given in (10.4), (10.8) and (10.20), respectively. Using this explicit
expression for $\mathbf{L}_{1}$, we see that

$(\mathbf{L}_{1}(s))_{21}=\frac{\left({\mathbf{\Phi}}^{(1)}_{n}\right)_{21}}{2\mathsf{m}(s)^{1/2}\psi(s)^{1/2}}.$

Both functions $\mathsf{m}$ and $\psi$ are analytic in a neighborhood of the
origin, and vanish linearly therein. Combining in addition with (8.5), we see
that the product $\mathsf{m}(z)^{1/2}\psi(z)^{1/2}$ admits an analytic
continuation near the origin, with the expansion

$\mathsf{m}(s)^{1/2}\psi(s)^{1/2}=\frac{\mathsf{c}_{V}}{a}s(1+\mathcal{O}(s)),\quad
s\to 0.$

Thus, computing residues

$\int_{\partial
U_{0}}(\mathbf{L}_{1}(s))_{21}\mathrm{d}s=-\pi\mathrm{i}\frac{a}{\mathsf{c}_{V}}\left({\mathbf{\Phi}}^{(1)}_{n}\right)_{21}.$

Similarly, using now (10.8) we obtain

$\int_{\partial_{U_{0}}}\frac{\mathrm{i}\mathsf{q}^{(1)}(s)(\mathsf{m}(s)-1)}{2\mathsf{m}(s)^{1/2}}\mathrm{d}s=-\mathsf{t}^{1/2}a^{1/2}F_{-1/2}(\mathsf{s})$

Hence,

$\left(\mathbf{R}_{1}\right)_{21}=\frac{1}{n^{1/3}}\left(\frac{a}{2\mathsf{c}_{V}}\left(\mathbf{\Phi}_{n}^{(1)}\right)_{21}+\frac{\mathsf{t}^{1/2}a^{1/2}}{4\pi\mathrm{i}}F_{-1/2}(\mathsf{s})\right)+\mathcal{O}(n^{-2/3}),$

and (11.4) updates to

$\left(\upgamma_{n-1}^{(n,Q)}(\mathsf{s})\right)^{2}=\operatorname{\rm
e}^{-2n\ell_{V}}\left(\frac{a}{8\pi}-\frac{1}{n^{1/3}}\frac{a}{4\pi\mathrm{i}}\left(\mathbf{\Phi}_{n}^{(1)}\right)_{21}+\mathcal{O}(n^{-2/3})\right),\quad
n\to\infty,$

uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$. We now need only to apply
(10.18) with any $\nu\geq 1/3$ to complete the proof.

###  11.3. Proof of Theorem 2.2

Unlike the already proven major results, the proof of Theorem 2.2 is not a
straightforward consequence of the steepest descent analysis concluded with
Theorem 10.7. It does rely substantially on Theorem 10.7, but several other
inputs are also needed along the way. Equipped with (9.10), the idea is to
account for the different approximations for $\mathsf{K}_{n}^{Q}$ on the
different components

$\mathbb{R}\setminus(U_{0}\cup D_{\delta}(-a)\cup(-a,0)),\quad\mathbb{R}\cup
U_{0},\quad(-a,-\delta,-a+\delta)\quad\text{and}\quad(-a+\delta,0)\setminus
U_{0}$

that arise from the RHP, and integrate each such approximation. With
Proposition 9.1 in mind, we are thus able to recover asymptotics for
$\mathsf{L}_{n}^{Q}$ itself. As one would expect, it turns out that in this
process the terms that arise away from $U_{0}$ become all exponentially
negligible, and only the contribution from $U_{0}$ survives in the leading
contribution. The contribution that arrives this way involves
$\mathbf{\Phi}_{n}$, and we further need to split it into different parts and
still account for some exact cancellations to arrive at the leading asymptotic
contribution. We postpone this analysis to the next section, where it is split
into several different lemmas, and summarize the outcome with the next result.
For its statement, we recall that $\mathsf{h}_{n}$ and
$\widehat{\mathbf{\Phi}}_{n}$ were introduced in (10.16), and we also set

$\widehat{\mathbf{\Delta}}_{n}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{I}+(1+\operatorname{\rm
e}^{-\mathsf{h}_{n}(u)})\chi_{(0,+\infty)(x)}\mathbf{E}_{21},\quad
x\in\mathbb{R}.$ (11.5)

######  Proposition 11.1.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$. There exists a function
$\mathsf{R}(u)=\mathsf{R}_{1}(u\mid\mathsf{t})$ satisfying

$\int_{\mathsf{s}}^{\infty}\mathsf{R}(u)\mathrm{d}u=\mathcal{O}(n^{-1/3}),\quad
n\to\infty,$

uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$, and for which the
identity

$\int_{-\infty}^{\infty}\mathsf{K}_{n}^{Q}(x,x\mid u)\frac{\omega_{n}(x\mid
u)}{1+\operatorname{\rm e}^{u+n^{2/3}Q(x)}}\mathrm{d}x\\\
=\frac{1}{2\pi\mathrm{i}}\int_{n^{2/3}\zeta_{0}}^{n^{2/3}\zeta_{1}}\frac{\operatorname{\rm
e}^{\mathsf{h}_{n}(u)}}{\left(1+\operatorname{\rm
e}^{\mathsf{h}_{n}(u)}\right)^{2}}\left[\widehat{\mathbf{\Delta}}_{n}(x)^{-1}\widehat{\mathbf{\Phi}}_{n,+}(x)^{-1}\left(\widehat{\mathbf{\Phi}}_{n,+}\widehat{\mathbf{\Delta}}_{n}\right)^{\prime}(x)\right]_{21}\mathrm{d}x+\mathsf{R}(u),$

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

In words, Proposition 11.1 is saying that the major contribution to
$\mathsf{L}_{n}^{Q}$ comes from the neighborhood $U_{0}$, so from the local
parametrix $\widehat{\mathbf{\Phi}}_{n}$. But according to the developments
from the previous sections, this local parametrix is close to the id-PII
parametrix ${\mathbf{\Phi}}_{0}$, and we are ready to conclude our last major
result.

###### Proof of Theorem 2.2.

We combine Propositions 9.1 and (11.1) to obtain

$\log\mathsf{L}_{n}^{Q}(\mathsf{s})=-\frac{1}{2\pi\mathrm{i}}\int_{\mathsf{s}}^{\infty}\int_{n^{2/3}\zeta_{0}}^{n^{2/3}\zeta_{1}}\frac{\operatorname{\rm
e}^{\mathsf{h}_{n}(u)}}{\left(1+\operatorname{\rm
e}^{\mathsf{h}_{n}(u)}\right)^{2}}\left[\widehat{\mathbf{\Delta}}_{n}(x)^{-1}\widehat{\mathbf{\Phi}}_{n,+}(x)^{-1}\left(\widehat{\mathbf{\Phi}}_{n,+}\widehat{\mathbf{\Delta}}_{n}\right)^{\prime}(x)\right]_{21}\mathrm{d}x\mathrm{d}u+\mathcal{O}(n^{-1/3}),$

valid as $n\to\infty$ and uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$. Next, with the
identifications (10.16)–(10.17) in mind, we estimate the integral on the
right-hand side with the help of Theorem 7.2 and conclude the proof. ∎

##  12\. Technical Lemmas

It remains to prove Proposition 11.1, analysis which we split into several
technical lemmas in this section.

Our starting point is the integral representation for $\mathsf{L}_{n}^{Q}$
from Proposition 9.1 with the asymptotic information for $\mathsf{K}_{n}^{Q}$
provided by the RHP analysis. For $x\in\mathbb{R}$, set

$\mathsf{A}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left[\mathbf{\Lambda}_{n}(x)^{-1}\mathbf{P}_{+}(x)^{-1}\mathbf{R}_{+}(x)^{-1}\mathbf{R}_{+}^{\prime}(x)\mathbf{P}_{+}(x)\mathbf{\Lambda}_{n}(x)+\mathbf{\Lambda}_{n}(x)^{-1}\mathbf{P}_{+}(x)^{-1}\mathbf{P}_{+}^{\prime}(x)\mathbf{\Lambda}_{n}(x)\right]_{21}.$
(12.1)

Recalling (9.11), the unwrap of the transformations of the RHP yields the
identity

$\mathsf{K}_{n}^{Q}(x,x\mid u)\frac{\omega_{n}(x\mid u)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}=\frac{\sigma_{n}(x\mid u)\operatorname{\rm
e}^{-2n\phi_{+}(x)}}{2\pi\mathrm{i}(1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)})}\left(\mathsf{A}(x)+\left(\frac{\operatorname{\rm
e}^{2n\phi_{+}(x)}}{\sigma_{n}(x)}\right)^{\prime}\chi_{\mathcal{G}^{+}}(x)\right)$

We now need to integrate each of the terms on the right-hand side above, first
in $x\in\mathbb{R}$ and then in $u\in(\mathsf{s},+\infty)$. Each term will be
analyzed individually, also depending on whether we integrate $x$ in the bulk,
each of the edges or away from the support $[-a,0]$ of the equilibrium
measure. To simplify notation, it is convenient to introduce the additional
notation for each relevant integral, and for an arbitrary set
$J\subset\mathbb{R}$ denote

$\mathsf{I}_{1}(J)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{J}\frac{\sigma_{n}(x\mid
u)\operatorname{\rm e}^{-2n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathsf{A}(x)\mathrm{d}x$ (12.2)

and

$\displaystyle\mathsf{I}_{2}(J)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\int_{J}\frac{\sigma_{n}(x\mid u)\operatorname{\rm
e}^{-2n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\left(\frac{\operatorname{\rm
e}^{2n\phi_{+}(x)}}{\sigma_{n}(x)}\right)^{\prime}\mathrm{d}x$ (12.3)
$\displaystyle=$
$\displaystyle\int_{J}\left(2n\phi_{n,+}(x)-\frac{n^{2/3}Q^{\prime}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\right)\frac{1}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x,$

which are functions of $u\in\mathbb{R}$ as well. With
$\varepsilon_{0},\varepsilon_{1}>0$ being determined by
$(-\varepsilon_{0},\varepsilon_{1})=U_{0}\cap\mathbb{R}$, the split

$2\pi\mathrm{i}\int_{-\infty}^{\infty}\mathsf{K}_{n}^{Q}(x,x\mid
u)\frac{\omega_{n}(x\mid u)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x=\mathsf{I}_{1}(-\infty,-a-\delta)+\mathsf{I}_{1}(-a-\delta,a+\delta)\\\
+\mathsf{I}_{1}(a+\delta,-\varepsilon_{0})+\mathsf{I}_{1}(-\varepsilon_{0},\varepsilon_{1})+\mathsf{I}_{1}(\varepsilon_{1},+\infty)+\mathsf{I}_{2}(-a,-\varepsilon_{0})+\mathsf{I}_{2}(-\varepsilon_{0},0)$
(12.4)

is immediate, and with the next series of lemmas we estimate each of the terms
on the right-hand side.

######  Lemma 12.1.

Fix $\mathsf{s}\in\mathbb{R}$ and $\mathsf{t}_{0}\in(0,1)$. There exists
$\eta>0$ for which the estimate

$\mathsf{I}_{1}(-\infty,-a-\delta)+\mathsf{I}_{1}(-a+\delta,-\varepsilon_{0})+\mathsf{I}_{2}(-a,-\varepsilon_{0})=\mathcal{O}\left(\operatorname{\rm
e}^{-u}\operatorname{\rm e}^{-n^{2/3}\eta}\right)$

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

###### Proof.

On the intervals $(-\infty,-a-\delta)$ and $(\varepsilon_{1},\infty)$ the
function $\mathbf{\Lambda}_{n}$ is identically the identity matrix. On the
interval $(a+\delta,-\varepsilon_{0})$ the nontrivial entry of
$\mathbf{\Lambda}_{n}$ is $\operatorname{\rm e}^{2n\phi_{+}}/\sigma_{n}$ and,
because $Q<0$ and $\phi_{+}$ is purely imaginary in this interval, this
quotient is bounded. Also, away from the endpoints $-a$ and $0$ we have
$\mathbf{P}\equiv\mathbf{G}$. Both $\mathbf{P}$ and $\mathbf{G}$, and their
$x$-derivatives, decay as $x\to\pm\infty$ and remain bounded as $n\to\infty$,
all uniformly in $u$ and $\mathsf{t}$ as claimed. All of these facts combined
together, we obtain that for some constants $K>0$ and $n_{0}\geq 1$

$|\mathsf{A}(x)|\leq K,\quad\text{for
all}\;x\in\mathbb{R}\setminus\left((-a-\delta,-a+\delta)\cup(-\varepsilon_{0},\varepsilon_{1})\right),\;n\geq
n_{0},\;\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0},\;u\geq\mathsf{s}.$
(12.5)

Next, we now use that $Q\geq 0$ on the interval $(-\infty,0)$ to bound

$0\leq\frac{\sigma_{n}(x\mid u)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\leq\frac{1}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\leq\operatorname{\rm e}^{-u},\quad x\leq 0,$ (12.6)

which is valid for any $n\geq 1,u\in\mathbb{R},\mathsf{t}>0$. Thus, combining
everything we obtain

$|\mathsf{I}_{1}(-\infty,-a-\delta)|\leq K\operatorname{\rm
e}^{-u}\int_{-\infty}^{-a-\delta}\operatorname{\rm
e}^{-2n\phi_{+}(x)}\mathrm{d}x,$

and using Proposition 8.1–(iv),(v), this proves the bound for
$\mathsf{I}_{1}(-\infty,-a-\delta)$.

For the second integral, we term $\phi_{+}$ is oscillatory, so to obtain the
exponential decay in the $x$-integral we have to argue differently and as
follows. The function

$v\mapsto\frac{1}{1+\operatorname{\rm e}^{-v}}\frac{1}{1+\operatorname{\rm
e}^{v}}=\frac{\operatorname{\rm e}^{-v}}{(1+\operatorname{\rm e}^{-v})^{2}}$

is strictly increasing on $(-\infty,0)$ and strictly decreasing on
$(0,+\infty)$. Because $Q>0$ on $(-\infty,0)$ and $Q(0)=0$, by reducing
$U_{0}$ if necessary we can assume without loss of generality that $Q(x)\geq
Q(-\varepsilon_{0})$ for every $x\in[-a,-\varepsilon_{0}]$. This way,
$v=u+n^{2/3}Q(x)\geq
v_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=u+n^{2/3}Q(-\varepsilon_{0})$
and, because $u$ is assumed to be bounded from below, we can make sure that
$v_{0}>0$ for every $u$. Therefore

$0\leq\frac{\sigma_{n}(x\mid u)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\leq\frac{\operatorname{\rm
e}^{-v_{0}}}{(1+\operatorname{\rm e}^{-v_{0}})^{2}}\leq\operatorname{\rm
e}^{-u-n^{2/3}Q(-\varepsilon_{0})},$

and upon integration and using Proposition 8.1–(ii), we obtain

$|\mathsf{I}_{2}(-a+\delta,-\varepsilon_{0})|\leq(a-\delta+\varepsilon_{0})K\operatorname{\rm
e}^{-u-n^{2/3}Q(-\varepsilon_{0})}.$

for every $n\geq n_{0}$ and $\mathsf{t},u$ as claimed.

For the estimate for $\mathsf{I}_{2}(-a,-\varepsilon_{0})$ we use again the
last inequality in (12.6) and also that both $\phi^{\prime}_{+}$ and
$Q^{\prime}$ are continuous, and hence bounded, on $(-a,-\varepsilon_{0})$.
This concludes the proof. ∎

For the integral over $(\varepsilon_{1},+\infty)$, it is easier to actually
estimate its $u$-integral directly.

######  Lemma 12.2.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$. There exists $\eta>0$ for
which the estimate

$\int_{\mathsf{s}}^{\infty}\mathsf{I}_{1}(\varepsilon_{1},+\infty)\mathrm{d}u=\mathcal{O}\left(\operatorname{\rm
e}^{-n\eta}\right)$

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

###### Proof.

With the bound (12.5) we see that it is enough to estimate the integral

$\int_{\mathsf{s}}^{\infty}\int_{\varepsilon_{1}}^{\infty}\frac{\sigma_{n}(x\mid
u)\operatorname{\rm e}^{-2n\phi(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x\mathrm{d}u=\int_{\varepsilon_{1}}^{\infty}\frac{\operatorname{\rm
e}^{-2n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{\mathsf{s}+n^{2/3}Q(x)}}\mathrm{d}x,$

where for the equality we used Tonelli’s Theorem to interchange the order of
integration, and then integrated exactly. The term $(1+\operatorname{\rm
e}^{\mathsf{s}+n^{2/3}Q(x)})^{-1}$ is bounded by $1$, and using Proposition
8.1–(iv),(v) we see that the integral of $\operatorname{\rm e}^{-2n\phi}$ is
$\mathcal{O}(\operatorname{\rm e}^{-\eta n})$ for some $\eta>0$ independent of
$\mathsf{s}$ and $\mathsf{t}$, which concludes the proof. ∎

Next, we analyze the contribution coming from a neighborhood of the endpoint
$z=a$.

######  Lemma 12.3.

Fix $\mathsf{s}\in\mathbb{R}$, $\mathsf{t}_{0}\in(0,1)$. There exists $\eta>0$
such that the estimate

$\mathsf{I}_{1}(-a-\delta,-a+\delta)=\mathcal{O}(\operatorname{\rm e}^{-u-\eta
n^{2/3}}),\quad n\to\infty,$

is valid uniformly for $u\geq\mathsf{s}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$.

###### Proof.

On the neighborhood $D_{\delta}(-a)$ the function
$\mathbf{P}=\mathbf{P}^{(a)}$ is the Airy local parametrix (10.12), which
involves the function $\mathbf{\Phi}_{\operatorname{Ai}}$ evaluated at the
argument $\zeta=n^{2/3}\varphi(z)$. In the $\zeta$-plane, we fix $R>0$ for
which the asymptotic expansion (10.10) is valid for $|\zeta|>R$, and split the
analysis into two cases, namely for $|n^{2/3}\varphi(z)|\leq R$ and
$|n^{2/3}\varphi(z)|\geq R$, and for

$A_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{x\in(-a-\delta,-a+\delta)\mid|n^{2/3}\varphi(x)|\leq
R\\},\quad
B_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(-a-\delta,-a+\delta)\setminus
A_{n}$

we write

$\mathsf{I}_{1}(-a-\delta,-a+\delta)=\mathsf{I}_{1}(A_{n})+\mathsf{I}_{1}(B_{n}).$

and now analyze each integral on the right-hand side separately.

For $|n^{2/3}\varphi(z)|\leq R$, the terms
$\mathbf{\Phi}_{\operatorname{Ai},+}(\zeta=n^{2/3}\varphi(z))$ and
$\mathbf{\Phi}^{\prime}_{\operatorname{Ai},+}(\zeta=n^{2/3}\varphi(z))$
consist of continuous functions evaluated inside the compact interval
$[-R,R]$, and therefore they are bounded. By the same reason, the expression
(10.11) shows that $\mathbf{F}$ is bounded for $|n^{2/3}\varphi(z)|\leq R$. On
the other hand, without further analysis we obtain that $\mathbf{F}^{\prime}$
may grow at most as $\mathcal{O}(n^{1/6})$. Finally, combining with (10.9) and
the fact that the determinant of $\mathbf{P}$ is identically $1$, we conclude
that for $|n^{2/3}\varphi(x)|\leq R$,

$\mathbf{P}_{+}(x)=\mathcal{O}(1)\operatorname{\rm
e}^{n\phi_{+}(x)\bm{\sigma}_{3}},\quad\mathbf{P}_{+}(x)^{-1}=\operatorname{\rm
e}^{-n\phi_{+}(x)\bm{\sigma}_{3}}\mathcal{O}(1)\quad\text{and}\quad\mathbf{P}_{+}^{\prime}(x)=\mathcal{O}(n)\operatorname{\rm
e}^{-n\phi_{+}(x)\bm{\sigma}_{3}}$ (12.7)

which is valid as $n\to\infty$ and uniformly for $x\in(-a-\delta,-a+\delta)$,
also uniformly for $u\geq\mathsf{s}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$.

We use (12.7) back into (12.1), and combined with the fact that
$|\sigma_{n}^{-1}|\leq 2$ for $x\leq(-\infty,0)$ the result is that

$\mathsf{A}(x)=\left[\mathbf{\Lambda}_{n}(x)^{-1}\operatorname{\rm
e}^{-n\phi_{+}(x)\bm{\sigma}_{3}}\mathcal{O}(n)\operatorname{\rm
e}^{n\phi_{+}(x)\bm{\sigma}_{3}}\mathbf{\Lambda}_{n}(x)\right]_{21}=\operatorname{\rm
e}^{2n\phi_{+}(x)}\mathcal{O}(n),\quad n\to\infty,$

where the last error term is a scalar error, valid uniformly in
$x,u,\mathsf{t}$ as before. Integrating, we obtain that for some absolute
constant $K>0$,

$|\mathsf{I}_{1}(A_{n})|\leq Kn\int_{A_{n}}\frac{\sigma_{n}(x\mid
u)}{1+\operatorname{\rm e}^{u+n^{2/3}Q(x)}}\mathrm{d}x\leq
Kn\int_{A_{n}}\operatorname{\rm
e}^{-u-n^{2/3}Q(x)}\mathrm{d}x=\mathcal{O}(\operatorname{\rm e}^{-u-\eta
n^{2/3}}),\quad n\to\infty,$ (12.8)

where for the second inequality we used that $0<\sigma_{n}\leq 1$ and for the
last estimate we used that $Q$ is strictly positive on
$(-a-\delta,-a+\delta)\supset A_{n}$.

Next, we consider the case $|n^{2/3}\varphi(z)|\geq R$. In such situation, the
asymptotics (10.10) take place, and when combined with (10.11) they yield

$\mathbf{P}_{+}(x)=\mathbf{G}_{+}(x)\operatorname{\rm
e}^{-\bm{\sigma}_{3}\log\sigma_{n}(x)/2}\left(\mathbf{I}+\mathcal{O}(n^{-1})\right)\operatorname{\rm
e}^{\bm{\sigma}_{3}\log\sigma_{n}(x)/2}=\mathbf{G}_{+}(x)\left(\mathbf{I}+\mathcal{O}(n^{-1})\right).$
(12.9)

where for the last equality we used (10.9).

The matrix $\mathbf{P}_{+}$ is bounded as $x\to 0$, whereas $\mathbf{G}$ is
not, but the cancellation that leads to this boundedness of $\mathbf{P}$ is
not captured by this asymptotics. Nevertheless, as we need some uniform
control over $x$, we now account for the behavior of $x\to 0$ in a rough
manner as follows. We are assuming that $|n^{2/3}\varphi(x)|\geq R$, and
because $\varphi$ is conformal with $\varphi(0)=0$ this means that $|x+a|\geq
c/n^{2/3}$, for some fixed $c>0$. The function $\mathbf{G}$ has a fourth-root
singularity at $x=0$, and therefore we arrive at the rough estimate

$\mathbf{P}_{+}(x)=\mathcal{O}(n^{1/6}),\quad n\to\infty.$ (12.10)

The estimate (12.9) can be differentiated term by term. With arguments similar
to the ones we just applied, we arrive at the rough estimate

$\mathbf{P}^{\prime}_{+}(x)=\mathcal{O}(n^{7/6}),\quad n\to\infty.$ (12.11)

These latter two estimates are valid uniformly for $u\geq\mathsf{s}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$ as $n\to\infty$.

Finally, on the interval $(-a,0)$ the factor $\phi_{+}$ is purely imaginary,
implying that $\mathbf{\Lambda}_{n}^{\pm 1}$ remains bounded therein. All
combined, we obtained that

$\mathsf{A}(x)=\mathcal{O}(n),\quad n\to\infty,$

uniformly for $x\in B_{n}$ and $u,\mathsf{t}$ as claimed. Proceeding as in
(12.8) we obtain a bound for $I_{1}(B_{n})$ and complete the proof. ∎

It remains to analyze the two integrals
$\mathsf{I}_{1}(-\varepsilon_{0},\varepsilon_{1})$ and
$\mathsf{I}_{2}(-\varepsilon_{0},0)$. The hardest analysis is the integral
$\mathsf{I}_{1}$ which, as will turn out, contains both the leading
contribution, a term to cancel $\mathsf{I}_{2}$ and asymptotically negligible
terms. For ease of presentation, we explore the expression for $\mathsf{A}$ in
(12.1) as a sum and split

$\mathsf{I}_{1}(-\varepsilon_{0},\varepsilon_{1})=\mathsf{J}_{1}(-\varepsilon_{0},\varepsilon_{1})+\mathsf{J}_{2}(-\varepsilon_{0},\varepsilon_{1}),$
(12.12)

where, for any measurable set $J\subset\mathbb{R}$,

$\displaystyle\mathsf{J}_{1}(J)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{J}\left[\mathbf{\Lambda}_{n}(x)^{-1}\mathbf{P}_{+}(x)^{-1}\mathbf{R}(x)^{-1}\mathbf{R}^{\prime}(x)\mathbf{P}_{+}(x)\mathbf{\Lambda}_{n}(x)\right]_{21}\frac{\sigma_{n}(x\mid
u)\operatorname{\rm e}^{-2n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x,$ (12.13) $\displaystyle\mathsf{J}_{2}(J)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{J}\left[\mathbf{\Lambda}_{n}(x)^{-1}\mathbf{P}_{+}(x)^{-1}\mathbf{P}_{+}^{\prime}(x)\mathbf{\Lambda}_{n}(x)\right]_{21}\frac{\sigma_{n}(x\mid
u)\operatorname{\rm e}^{-2n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x,$

and analyze each of these terms separately. For the estimation of
$\mathsf{J}_{1}$, it is also easier to perform the $u$-integral, just as we
did in Lemma 12.2.

######  Lemma 12.4.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$. The estimate

$\int_{\mathsf{s}}^{\infty}\mathsf{J}_{1}(-\varepsilon_{0},\varepsilon_{1})\mathrm{d}u=\mathcal{O}(n^{-1/3})$
(12.14)

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

###### Proof.

The idea is similar to the proof of Lemma 12.3. We fix a number $R>0$ for
which the asymptotic expansion (4.3) for
$\mathbf{\Phi}=\widehat{\mathbf{\Phi}}_{n}$ is valid for $|\zeta|\geq R$,
uniformly in $\mathsf{t},\mathsf{s}$ as claimed (see Remark 7.8). Then,
introduce

$C_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{x\in(-\varepsilon_{0},\varepsilon_{1})\mid|n^{2/3}\psi(x)|\leq
R\\},\quad
D_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(-\varepsilon_{0},\varepsilon_{1})\setminus
C_{n}.$ (12.15)

We now find bounds for the integrands, with separate arguments for each of
$C_{n}$ and $D_{n}$.

Recalling (10.22), on the interval $(-\varepsilon_{0},\varepsilon_{1})$ the
parametrix is
$\mathbf{P}=\mathbf{P}^{(0)}=\mathbf{F}_{n}\mathbf{\Phi}_{n}\operatorname{\rm
e}^{n\phi\bm{\sigma}_{3}}$, with $\mathbf{F}_{n}$ and $\mathbf{\Phi}_{n}$ as
in (10.16). Using the definition of $\mathbf{F}_{n}$ in (10.16) and Theorem
10.7, we express

$\mathbf{F}_{n}^{-1}\mathbf{R}^{-1}\mathbf{R}^{\prime}\mathbf{F}_{n}=\mathbf{F}_{n}^{-1}(\mathbf{R}^{-1}-\mathbf{I})\mathbf{R}^{\prime}\mathbf{F}_{n}-\mathbf{F}_{n}^{-1}\mathbf{R}^{\prime}\mathbf{F}_{n}=-\mathbf{F}_{n}^{-1}\mathbf{R}^{\prime}\mathbf{F}_{n}+\mathcal{O}(n^{-1/3}),\quad
n\to\infty,$ (12.16)

valid uniformly when evaluated at $x\in C_{n}$ and also uniformly for
$\mathsf{s},\mathsf{t}$ as claimed. Using again the definition of
$\mathbf{F}_{n}$, the fact that $\psi/\mathsf{m}$ remains bounded near $z=0$
and (10.30),

$\mathbf{F}_{n}^{-1}\mathbf{R}^{\prime}\mathbf{F}_{n}=\frac{1}{n^{1/3}}\psi^{\bm{\sigma}_{3}/4}\mathsf{m}^{-\bm{\sigma}_{3}/4}n^{\bm{\sigma}_{3}/6}\widehat{\mathbf{R}}_{1}^{\prime}n^{-\bm{\sigma}_{3}/4}\mathsf{m}^{\bm{\sigma}_{3}/4}\psi^{-\bm{\sigma}_{3}/4}+\mathcal{O}(n^{-1/3}).$

A careful inspection on (10.29) shows that
$(\widehat{\mathbf{R}}_{1}(z))_{12}$ is independent of $z$, so $R^{\prime}$
has zero $(1,2)$-entry. Therefore we conclude that

$\mathbf{F}_{n}^{-1}\mathbf{R}^{-1}\mathbf{R}^{\prime}\mathbf{F}_{n}=\mathcal{O}(n^{-1/3}),$

stressing that this is valid for $x\in C_{n}$. Also, from (10.19) we learn
that $\mathbf{\Phi}_{n}(x)=\mathcal{O}(1)$ uniformly on $C_{n}$. Everything we
have so far combined yields that

$\left[\mathbf{\Lambda}_{n}^{-1}\mathbf{P}_{+}^{-1}\mathbf{R}^{-1}\mathbf{R}^{\prime}\mathbf{P}\mathbf{\Lambda}_{n}\right]_{21}=\left[\mathbf{\Lambda}_{n}^{-1}\operatorname{\rm
e}^{-n\phi\bm{\sigma}_{3}}\mathcal{O}(n^{-1/3})\operatorname{\rm
e}^{n\phi_{3}\bm{\sigma}_{3}}\mathbf{\Lambda}_{n}\right]_{21}=\operatorname{\rm
e}^{-2n\phi}\left[\mathbf{\Delta}_{n}^{-1}\mathcal{O}(n^{-1/3})\mathbf{\Delta}_{n}\right]_{21}.$

Along $(-a,0)$ we have $Q>0$ so that $1/\sigma_{n}$ remains bounded uniformly,
and consequently $\mathbf{\Delta}_{n}=\mathcal{O}(1)$ in the same interval.
Using this information in the last displayed equation, we thus obtain that

$\mathsf{J}_{1}(C_{n})=\int_{C_{n}}\mathcal{O}(n^{-1/3})\frac{\sigma_{n}(x\mid
u)}{1+\operatorname{\rm e}^{u+n^{2/3}Q(x)}}\mathrm{d}x,\quad n\to\infty,$
(12.17)

uniformly for $u\geq\mathsf{s}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$, and where the error term is uniform for $x\in C_{n}$.

Next, along $D_{n}$ we now use the asymptotic expansion (4.3) for
$\mathbf{\Phi}=\widehat{\mathbf{\Phi}}_{n}$ and the definition of
$\mathsf{F}_{n}$ and obtain that

$\mathbf{P}(x)=\mathbf{U}_{0}\mathsf{m}_{+}(x)^{\bm{\sigma}_{3}/4}\mathbf{U}_{0}^{-1}\left(1+\mathcal{O}(n^{-1/3})\right)=\mathbf{U}_{0}\begin{pmatrix}\mathcal{O}(1)&0\\\
0&\mathcal{O}(n^{1/6})\end{pmatrix}\mathbf{U}_{0}^{-1}\left(1+\mathcal{O}(n^{-1/3})\right),\quad
x\in D_{n},$

and similarly

$\mathbf{P}(x)=\left(1+\mathcal{O}(n^{-1/3})\right)\mathbf{U}_{0}\begin{pmatrix}\mathcal{O}(n^{1/6})&0\\\
0&\mathcal{O}(1)\end{pmatrix}\mathbf{U}_{0}^{-1},\quad x\in D_{n},$

all valid as $n\to\infty$, uniformly for $x\in D_{n}$ and uniformly in
$u,\mathsf{t}$ as claimed

With the same arguments that we applied in (12.16), we obtain along $D_{n}$ as
well

$\mathbf{P}^{-1}\mathbf{R}^{-1}\mathbf{R}^{\prime}\mathbf{P}=\mathcal{O}(n^{-1/3}),\quad\text{so
that}\quad\mathsf{J}_{1}(D_{n})=\int_{D_{n}}\mathcal{O}(n^{-1/3})\frac{\sigma_{n}(x\mid
u)\operatorname{\rm e}^{-2n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x,\quad n\to\infty.$

The factor $\phi_{+}$ is purely imaginary on $(-a,0)$ and positive on
$[0,+\infty)$, so the term $\operatorname{\rm e}^{-2n\phi_{+}}$ in the
integrand above is bounded by a uniform constant. Combining with (12.17), we
conclude

$\mathsf{J}_{1}(-\varepsilon_{0},\varepsilon_{1})=\mathcal{O}(n^{-1/3})\int_{-\varepsilon_{0}}^{\varepsilon_{1}}\frac{\sigma_{n}(x\mid
u)}{1+\operatorname{\rm e}^{u+n^{2/3}Q(x)}}\mathrm{d}x$

Finally, we now integrate in $u$ and use Tonelli’s Theorem to interchange the
order of integration, obtaining just like in the proof of Lemma 12.2 that

$\int_{\mathsf{s}}^{\infty}\int_{-\varepsilon_{0}}^{\varepsilon_{1}}\frac{\sigma_{n}(x\mid
u)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x\mathrm{d}u=\int_{-\varepsilon_{0}}^{\varepsilon_{1}}\frac{1}{1+\operatorname{\rm
e}^{\mathsf{s}+n^{2/3}Q(x)}}\mathrm{d}x\leq\varepsilon_{1}+\varepsilon_{0},$

which concludes the proof. ∎

Next, in our pursuit of analyzing (12.4), the final missing piece of the
puzzle is the term $\mathsf{J}_{2}(-\varepsilon_{0},\varepsilon_{0})$ that
arises from $\mathsf{I}_{1}(-\varepsilon_{0},\varepsilon_{0})$. To state the
rigorous results, we recall once again that the conformal map $\psi$ was
introduced in Proposition 8.2, the function $\mathsf{h}_{n}$ is given in
(10.16), the function $\widehat{\mathbf{\Delta}}_{n}$ is in (11.5) and in
addition also set

$\zeta_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\psi(-\varepsilon_{0})<0,\quad\zeta_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\psi(\varepsilon_{1})>0.$

######  Lemma 12.5.

Fix $\mathsf{s}_{0}>0$ and $\mathsf{t}_{0}\in(0,1)$. There exists a function
$\mathsf{R}_{1}(u)=\mathsf{R}_{n}(u\mid\mathsf{t})$ satisfying the estimate

$\int_{\mathsf{s}}^{\infty}\mathsf{R}(u)\mathrm{d}u=\mathcal{O}(n^{-2/3}),\quad
n\to\infty,$

uniformly for $\mathsf{s}\geq-\mathsf{s}_{0}$ and
$\mathsf{t}_{0}\leq\mathsf{t}\leq 1/\mathsf{t}_{0}$, and for which the
identity

$\mathsf{J}_{2}(-\varepsilon_{0},\varepsilon_{0})=-\mathsf{I}_{2}(-\varepsilon_{0},0)\\\
+\int_{n^{2/3}\zeta_{0}}^{n^{2/3}\zeta_{1}}\frac{\operatorname{\rm
e}^{\mathsf{h}_{n}(u)}}{\left(1+\operatorname{\rm
e}^{\mathsf{h}_{n}(u)}\right)^{2}}\left[\widehat{\mathbf{\Delta}}_{n}(x)^{-1}\widehat{\mathbf{\Phi}}_{n,+}(x)^{-1}\left(\widehat{\mathbf{\Phi}}_{n,+}\widehat{\mathbf{\Delta}}_{n}\right)^{\prime}(x)\right]_{21}\mathrm{d}x+\mathsf{R}_{1}(u),$
(12.18)

holds true for every $u\in\mathbb{R}$ and every $\mathsf{t}>0$.

###### Proof.

Recall that $\mathsf{J}_{2}$ was introduced in (12.13). In the interval
$(-\varepsilon_{0},\varepsilon_{1})$ the local parametrix
$\mathbf{P}=\mathbf{P}^{(0)}$ coincides with (10.22). A direct calculation
from (10.16) shows that the matrix $\mathbf{F}_{n}$ therein satisfies the
identity

$\mathbf{F}_{n}^{\prime}(z)=\frac{1}{4}\left(\frac{\mathsf{m}^{\prime}(z)}{\mathsf{m}(z)}-\frac{\psi^{\prime}(z)}{\psi(z)}\right)\mathbf{F}_{n}(z)\bm{\sigma}_{3}.$

With the help of this identity, we express

$\mathbf{P}_{+}^{-1}\mathbf{P}_{+}^{\prime}=\operatorname{\rm
e}^{-n\phi_{+}\bm{\sigma}_{3}}\left[\frac{1}{4}\left(\frac{\mathsf{m}^{\prime}}{\mathsf{m}}-\frac{\psi^{\prime}}{\psi}\right)\mathbf{\Phi}_{n,+}^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}+\mathbf{\Phi}_{n,+}^{-1}\mathbf{\Phi}_{n,+}^{\prime}+n\phi^{\prime}_{+}\bm{\sigma}_{3}\right]\operatorname{\rm
e}^{n\phi_{+}\bm{\sigma}_{3}},$

where both sides are evaluated at $x\in(-\varepsilon_{0},\varepsilon_{1})$.
Thus,

$\Bigg{[}\mathbf{\Lambda}_{n}^{-1}\mathbf{P}_{+}^{-1}\mathbf{P}_{+}^{\prime}\mathbf{\Lambda}_{n}\Bigg{]}_{21}\\\
\begin{aligned} &=-2n\frac{\phi_{+}^{\prime}\chi_{0}\operatorname{\rm
e}^{2n\phi_{+}}}{\sigma_{n}}+\operatorname{\rm
e}^{2n\phi_{+}}\left[\mathbf{\Delta}_{n}\mathbf{\Phi}_{n,+}^{-1}\mathbf{\Phi}_{n,+}^{\prime}\mathbf{\Delta}_{n}\right]_{21}+\frac{\operatorname{\rm
e}^{2n\phi_{+}}}{4}\left(\frac{\mathsf{m}^{\prime}}{\mathsf{m}}-\frac{\psi^{\prime}}{\psi}\right)\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Phi}_{n,+}^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}\mathbf{\Delta}_{n}\right]_{21}\\\
&\begin{multlined}=-2n\frac{\phi_{+}^{\prime}\chi_{0}\operatorname{\rm
e}^{2n\phi_{+}}}{\sigma_{n}}-\operatorname{\rm
e}^{2n\phi_{+}}\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Delta}_{n}^{\prime}\right]_{21}+\operatorname{\rm
e}^{2n\phi_{+}}\left[\mathbf{\Delta}_{n}\mathbf{\Phi}_{n,+}^{-1}\left(\mathbf{\Phi}_{n,+}\mathbf{\Delta}_{n}\right)^{\prime}\right]_{21}\\\
\phantom{-2n\frac{\phi_{+}^{\prime}\chi_{0}\operatorname{\rm
e}^{2n\phi_{+}}}{\sigma_{n}}-\operatorname{\rm
e}^{2n\phi_{+}}\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Delta}_{n}^{\prime}\right]_{21}}+\frac{\operatorname{\rm
e}^{2n\phi_{+}}}{4}\left(\frac{\mathsf{m}^{\prime}}{\mathsf{m}}-\frac{\psi^{\prime}}{\psi}\right)\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Phi}_{n,+}^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}\mathbf{\Delta}_{n}\right]_{21}\end{multlined}=-2n\frac{\phi_{+}^{\prime}\chi_{0}\operatorname{\rm
e}^{2n\phi_{+}}}{\sigma_{n}}-\operatorname{\rm
e}^{2n\phi_{+}}\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Delta}_{n}^{\prime}\right]_{21}+\operatorname{\rm
e}^{2n\phi_{+}}\left[\mathbf{\Delta}_{n}\mathbf{\Phi}_{n,+}^{-1}\left(\mathbf{\Phi}_{n,+}\mathbf{\Delta}_{n}\right)^{\prime}\right]_{21}\\\
\phantom{-2n\frac{\phi_{+}^{\prime}\chi_{0}\operatorname{\rm
e}^{2n\phi_{+}}}{\sigma_{n}}-\operatorname{\rm
e}^{2n\phi_{+}}\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Delta}_{n}^{\prime}\right]_{21}}+\frac{\operatorname{\rm
e}^{2n\phi_{+}}}{4}\left(\frac{\mathsf{m}^{\prime}}{\mathsf{m}}-\frac{\psi^{\prime}}{\psi}\right)\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Phi}_{n,+}^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}\mathbf{\Delta}_{n}\right]_{21}\end{aligned}$
(12.19)

We now multiply this last expression by $\operatorname{\rm
e}^{-2n\phi_{+}}\sigma_{n}/(1+\operatorname{\rm e}^{u+n^{2/3}Q})$ and
integrate. A direct calculation gives

$\frac{\sigma_{n}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q}}\left(2n\frac{\phi_{+}^{\prime}\chi_{0}}{\sigma_{n}}+\left[\mathbf{\Delta}_{n}^{-1}\mathbf{\Delta}_{n}^{\prime}\right]_{21}\right)=\frac{\sigma_{n}\chi_{0}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q}}\left(2n\frac{\phi_{+}^{\prime}}{\sigma_{n}}+\left(\frac{1}{\sigma_{n}}\right)^{\prime}\right)=\frac{\chi_{0}\sigma_{n}\operatorname{\rm
e}^{-2n\phi_{+}}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q}}\left(\frac{\operatorname{\rm
e}^{2n\phi_{+}}}{\sigma_{n}}\right)^{\prime},$

which is the integrand of
$\mathsf{I}_{2}(-\varepsilon_{0},\varepsilon_{1})=\mathsf{I}_{2}(-\varepsilon_{0},0)$,
see (12.3). Thus, from (12.19) we obtain

$\mathsf{J}_{2}(-\varepsilon_{0},\varepsilon_{1})=-\mathsf{I}_{1}(-\varepsilon_{0},0)+\int_{-\varepsilon_{0}}^{\varepsilon_{1}}\frac{\sigma_{n}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\left[\mathbf{\Delta}_{n}(x)^{-1}\mathbf{\Phi}_{n,+}(x)^{-1}\left(\mathbf{\Phi}_{n,+}(x)\mathbf{\Delta}_{n}(x)\right)^{\prime}\right]_{21}\mathrm{d}x+\mathsf{R}_{1}(u)$
(12.20)

where we have set

$\mathsf{R}_{1}(u)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{4}\int_{-\varepsilon_{0}}^{\varepsilon_{1}}\left(\frac{\mathsf{m}^{\prime}(x)}{\mathsf{m}(x)}-\frac{\psi^{\prime}(x)}{\psi(x)}\right)\frac{\sigma_{n}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\left[\mathbf{\Delta}_{n}(x)^{-1}\mathbf{\Phi}_{n,+}(x)^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}(x)\mathbf{\Delta}_{n}(x)\right]_{21}\mathrm{d}x.$

It is worth mentioning that both $\mathsf{m}$ and $\psi$ have a simple zero at
$x=0$, so the first term in the integrand of $\mathsf{R}$ remains bounded near
$x=0$.

Recalling (10.16), the integrand written explicitly in (12.20) is of the form

$f_{1}(n^{2/3}\psi(x))\left[\mathbf{f}_{2}(n^{2/3}\psi(x))\left(\mathbf{f}_{3}(n^{2/3}\psi(x))\right)^{\prime}\right]_{21}=n^{2/3}\psi^{\prime}(x)f_{1}(n^{2/3}\psi(x))\left[\mathbf{f}_{2}(n^{2/3}\psi(x))\mathbf{f}_{3}^{\prime}(n^{2/3}\psi(x))\right]_{21}$

with obvious choices of the functions $f_{1},\mathbf{f}_{2},\mathbf{f}_{3}$,
and performing the change of variables $u=n^{2/3}\psi(x)$ we obtain the
integral in the right-hand side of (12.18).

To conclude, it remains to show the bound for $\mathsf{R}$. For that, we use
again the sets $C_{n}$ and $D_{n}$ from (12.15) and analyze the integral over
each of these sets separately.

Along $C_{n}$, the convergence (10.19) ensures $\mathbf{\Phi}_{n,+}$ remains
bounded uniformly, and combined with the boundedness of all the other terms on
the whole interval $(-\varepsilon_{0},\varepsilon_{1})$ we obtain

$\int_{C_{n}}\left(\frac{\mathsf{m}^{\prime}(x)}{\mathsf{m}(x)}-\frac{\psi^{\prime}(x)}{\psi(x)}\right)\frac{\sigma_{n}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\left[\mathbf{\Delta}_{n}(x)^{-1}\mathbf{\Phi}_{n,+}(x)^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}(x)\mathbf{\Delta}_{n}(x)\right]_{21}\mathrm{d}x\\\
=\mathcal{O}(1)\int_{C_{n}}\frac{\sigma_{n}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x.$

The function $\psi$ is conformal, and consequently $|z|\leq cn^{-2/3}$ for
$z\in C_{n}$ and some constant $c$ independent of $\mathsf{t},u,n$. In
particular, this ensures that $n^{2/3}Q(x)\leq-\tilde{c}$ for every $x\in
C_{n}$ and a constant $\tilde{c}>0$, and therefore

$\frac{\sigma_{n}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\leq\frac{1}{1+\operatorname{\rm
e}^{u-\tilde{c}}}\leq\operatorname{\rm e}^{-u+\tilde{c}}.$

Still because $\psi$ is conformal, we are sure that the Lebesgue measure of
$C_{n}$ is $\mathcal{O}(n^{-2/3})$. Everything combined, we conclude the
estimate

$\int_{C_{n}}\left(\frac{\mathsf{m}^{\prime}(x)}{\mathsf{m}(x)}-\frac{\psi^{\prime}(x)}{\psi(x)}\right)\frac{\sigma_{n}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\left[\mathbf{\Delta}_{n}(x)^{-1}\mathbf{\Phi}_{n,+}(x)^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}(x)\mathbf{\Delta}_{n}(x)\right]_{21}\mathrm{d}x=\mathcal{O}(\operatorname{\rm
e}^{-u}n^{-2/3}),$

as $n\to\infty$, which is valid uniformly for $u,\mathsf{t}$ as claimed by the
Lemma.

Finally, on $D_{n}$ we use the expansion (5.10) for
$\mathbf{\Phi}=\mathbf{\Phi}_{n}$, which provides

$\mathbf{\Phi}_{n,+}(x)^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}(x)=\operatorname{\rm
e}^{n\phi_{+}(x)\bm{\sigma}_{3}}\left(\bm{\sigma}_{2}+\mathcal{O}(n^{-1/3})\right)\operatorname{\rm
e}^{-n\phi_{+}(x)\bm{\sigma}_{3}},\quad n\to\infty,$

which is valid uniformly for $x\in D_{n}$ and uniformly in the parameters
$u,\mathsf{t}$ as required. After some straightforward calculations, we thus
arrive at

$\int_{D_{n}}\left(\frac{\mathsf{m}^{\prime}(x)}{\mathsf{m}(x)}-\frac{\psi^{\prime}(x)}{\psi(x)}\right)\frac{\sigma_{n}(x)}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\left[\mathbf{\Delta}_{n}(x)^{-1}\mathbf{\Phi}_{n,+}(x)^{-1}\bm{\sigma}_{3}\mathbf{\Phi}_{n,+}(x)\mathbf{\Delta}_{n}(x)\right]_{21}\mathrm{d}x\\\
=\int_{D_{n}}\left(\frac{\mathsf{m}^{\prime}(x)}{\mathsf{m}(x)}-\frac{\psi^{\prime}(x)}{\psi(x)}\right)\frac{\mathrm{i}\sigma_{n}(x)\operatorname{\rm
e}^{-2n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\left[1+\frac{\chi_{0}(x)}{\sigma_{n}(x)^{2}}+\mathcal{O}(n^{-1/3})\right]\mathrm{d}x,\quad
n\to\infty,$

with, as always, uniform error term in $x\in D_{n}$, $u,\mathsf{t}$. Each of
the terms $(\mathsf{m}^{\prime}/\mathsf{m}-\psi^{\prime}/\psi)$ and
$\chi_{0}/\sigma_{n}^{2}$ is bounded on $(-\varepsilon_{0},\varepsilon_{1})$,
so to bound the integral above it is enough to estimate

$\int_{-\varepsilon_{0}}^{\varepsilon_{1}}\frac{\sigma_{n}(x)\operatorname{\rm
e}^{-n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x=\int_{-\varepsilon_{0}}^{0}\frac{\sigma_{n}(x)\operatorname{\rm
e}^{-n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x+\int_{0}^{\varepsilon_{1}}\frac{\sigma_{n}(x)\operatorname{\rm
e}^{-n\phi(x)}}{1+\operatorname{\rm e}^{u+n^{2/3}Q(x)}}\mathrm{d}x.$

For the integral over $(-\varepsilon_{0},0)$, we know that
$\operatorname{Re}\phi_{+}=0$, and then using Tonelli’s Theorem to integrate
first in $u$ we obtain that

$\int_{\mathsf{s}}^{+\infty}\left|\int_{-\varepsilon_{0}}^{0}\frac{\sigma_{n}(x)\operatorname{\rm
e}^{-n\phi_{+}(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x\right|\mathrm{d}u\leq\int_{-\varepsilon_{0}}^{0}\frac{1}{1+\operatorname{\rm
e}^{\mathsf{s}+n^{2/3}Q(x)}}\mathrm{d}x.$

Changing variables $v=n^{2/3}Q$ in this last integral, it then follows that
the right-hand side above is $\mathcal{O}(n^{-2/3})$ uniformly for
$\mathsf{s}\geq-\mathsf{s}_{0}$ and $\mathsf{t}_{0}\leq\mathsf{t}\leq
1/\mathsf{t}_{0}$.

Finally, for the integral over $(0,\varepsilon_{1})$ we now have that
$\phi\geq 0$ in this interval, and it is independent of $u$, so once again
interchanging order of integration we obtain

$0\leq\int_{\mathsf{s}}^{\infty}\int_{0}^{\varepsilon_{1}}\frac{\sigma_{n}(x)\operatorname{\rm
e}^{-n\phi(x)}}{1+\operatorname{\rm
e}^{u+n^{2/3}Q(x)}}\mathrm{d}x\mathrm{d}u=\int_{0}^{\varepsilon_{1}}\frac{\operatorname{\rm
e}^{-n\phi(x)}}{1+\operatorname{\rm
e}^{\mathsf{s}+n^{2/3}Q(x)}}\mathrm{d}x\leq\int_{0}^{\varepsilon_{1}}\operatorname{\rm
e}^{-n\phi(x)}\mathrm{d}x,$

and now changing variables $v=n\phi$ (which is well defined in this interval
because of the local behavior (8.4)) we see that the integral on the right-
most side is $\mathcal{O}(n^{-1})$. This completes the proof. ∎

To conclude, it remains to prove Proposition 11.1.

###### Proof of Proposition 11.1.

Recalling (12.4) and (12.12), the result is an immediate consequence of Lemmas
12.1–12.5. ∎

##  Appendix A Laplace-type integrals

For smooth enough functions $f,g$, consider the Laplace-type integral

$F(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{0}^{a}g(x)\log(1+\operatorname{\rm
e}^{-y-tf(x)})\mathrm{d}x,\quad a\in(0,+\infty].$

We are interested in the asymptotic behavior of $F(t)$ as $t\to+\infty$ while
$y$ also scales with $t$. The first result we need is a version of the usual
Watson’s Lemma.

######  Lemma A.1.

Fix $f(x)\equiv x$. Suppose also that $g\in L^{1}(0,a)$, and for some
$\delta>0$ and $\kappa\in[0,1)$ it is of the form

$g(x)=\frac{\widetilde{g}(x)}{x^{\kappa}},\quad 0<x<\delta,$

where $\widetilde{g}$ is $C^{\infty}$ in a neighborhood of the origin. Assume
that

$y\geq-M,$ (A.1)

for some $M>0$ fixed. Then $F(t)$ admits an expansion of the form

$F(t)=\sum_{k=0}^{N}\frac{1}{t^{k+1-\kappa}}\widetilde{g}^{(k)}(0)F_{k-\kappa}(y)+\mathcal{O}(t^{-N-2+\kappa}),\quad
t\to\infty,$

for any $N\geq 1$, where the error term is uniform for $y$ satisfying (A.1),
and the coefficients $F_{\beta}(y)$ are given by

$F_{\beta}(y)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{0}^{\infty}v^{\beta}\log(1+\operatorname{\rm
e}^{-y-v})\mathrm{d}v,\quad\beta>-1.$ (A.2)

For $y\geq 0$ and $k\in\mathbb{Z},k\geq 1$, $F_{k}$ admits the alternative
expression in terms of the polylog function $\operatorname{Li}_{s}$,

$F_{k}(y)=-k\Gamma(k)\operatorname{Li}_{2+k}(-\operatorname{\rm e}^{-y}).$

###### Proof.

The proof follows along the lines of the proof of the classical Watson’s Lemma
for Laplace transforms, see for instance [54], so we go over it without much
detail.

First off, we fix $\delta>0$ for which $\widetilde{g}$ admits a Taylor
expansion of order $N$ in a neighborhood of the interval $(-\delta,\delta)$,
and write for some $\eta>0$,

$\int_{0}^{a}g(x)\log(1+\operatorname{\rm
e}^{-y-tx})\mathrm{d}x=\int_{0}^{\delta}g(x)\log(1+\operatorname{\rm
e}^{-y-tx})\mathrm{d}x+\mathcal{O}(\operatorname{\rm e}^{-t\eta}),\quad
t\to\infty,$

where the error term is obtained using that $g\in L^{1}(0,\infty)$ and that
the function $x\mapsto\log(1+\operatorname{\rm e}^{-y-tx})$ is increasing. The
value $\eta>0$ is independent of $y$, and the error is uniform for $y$
satisfying (A.1). For the integral on the right-hand side, we expand
$\widetilde{g}$ in Taylor series and obtain

$\int_{0}^{\delta}g(x)\log(1+\operatorname{\rm
e}^{-y-tx})\mathrm{d}x=\sum_{k=0}^{N}\frac{\widetilde{g}^{(k)}(0)}{k!}\int_{0}^{\delta}u^{k-\kappa}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u+R_{N+1}(t),$

where the remainder satisfies

$|R_{N+1}(t)|\leq\frac{1}{(N+1)!}\sup_{0\leq
x\leq\delta}|\widetilde{g}^{(N+1)}(x)|\int_{0}^{\delta}u^{N+1-\kappa}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u$

Performing the change of variables $u=tv$, we see that for $F_{\beta}(y)$ as
defined,

$\int_{0}^{\delta}u^{\beta}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u=\frac{F_{\beta}(y)}{t^{\beta+1}}-\frac{1}{t^{\beta+1}}\int_{t\delta}^{\infty}v^{\beta}\log(1+\operatorname{\rm
e}^{-y-v}).$

Using (A.1) we see that the last integral is again
$\mathcal{O}(\operatorname{\rm e}^{-\eta t})$ uniformly for $y$. Also, it is
immediate that under the restriction (A.1) each coefficient $F_{\beta}(y)$ is
bounded in $y$, and this concludes the proof of the asymptotic formula. The
alternative representation of $F_{k}(y)$ in terms of polylogs follows by
integration by parts. ∎

Next, we move to more general exponents.

######  Proposition A.2.

Suppose that $f$ is $C^{\infty}$ in a neighborhood of the origin, with a
unique global minimum on $[0,a]$ at $x=0$ with $f(0)=0$, $f^{\prime}(0)>0$,
and that $g$ is as in Lemma A.1. In addition, suppose that $y$ satisfies (A.1)
for some $M>0$ fixed. Then $F(t)$ admits an expansion of the form

$F(t)=\sum_{k=0}^{N}\frac{1}{t^{k+1-\kappa}}\widehat{g}^{(k)}(0)F_{k-\kappa}(y)+\mathcal{O}(t^{-N-2+\kappa}),\quad
t\to\infty,$

for any $N\geq 1$, where the error term is uniform for $y$ satisfying (A.1),
$F_{k}$’s are as in Lemma A.1 and the function $\widehat{g}$ is $C^{\infty}$
and satisfies the identity

$\widehat{g}(f(x))=f(x)^{\kappa}g(x),\quad|x|\text{ sufficiently small}.$

###### Proof.

Because $f$ is assumed to be $C^{\infty}$ and to have a unique global minimum
at $x=0$, by the Inverse Function Theorem it admits an inverse $f^{-1}$ in a
neighborhood of the origin. With the change of variables

$u=f(x),\quad\text{and setting}\quad
u_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f(\delta),\quad\widehat{g}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{x^{\kappa}\widetilde{g}(f^{-1}(x))}{(f^{-1}(x))^{\kappa}}=x^{\kappa}g(f^{-1}(x)),$

for $\delta>0$ sufficiently small the integral is turned into

$\displaystyle F(t)$
$\displaystyle=\int_{0}^{\delta}g(x)\log(1+\operatorname{\rm
e}^{-y-tf(x)})\mathrm{d}x+\int_{\delta}^{a}g(x)\log(1+\operatorname{\rm
e}^{-y-tf(x)})\mathrm{d}x$
$\displaystyle=\int_{0}^{u_{0}}\frac{1}{x^{\kappa}}\widehat{g}(x)\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}x+\mathcal{O}(\operatorname{\rm e}^{-\eta t}),$

for some $\eta>0$. Applying Lemma A.1 to the remaining integral concludes the
proof. ∎

Next we move to a different scaling regime on $y$.

######  Lemma A.3.

Let $f$ and $g$ be as in Proposition A.2. For any fixed constants $M>0$ and
$\alpha\in[0,1)$, assume that

$-Mt^{\alpha}\leq y\leq-\frac{1}{M}.$ (A.3)

Then $F(t)$ admits an expansion of the form

$F(t)=\sum_{k=0}^{N}\frac{1}{t^{k+1-\kappa}}\widehat{g}^{(k)}(0)\varphi_{k-\kappa}(y)+\mathcal{O}(t^{-N-2+\kappa}|y|^{N+3-\kappa}),\quad
t\to\infty,$ (A.4)

for any $N\geq 1$, where the error term is uniform for $y$ satisfying (A.3),
and the coefficients $\varphi_{k}(y)$ are of the form

$\displaystyle\varphi_{\beta}(y)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varphi^{(1)}_{\beta}(y)+\varphi^{(2)}_{\beta}(y)+\varphi^{(3)}_{\beta}(y),\quad\text{with}$
$\displaystyle\varphi_{\beta}^{(1)}(y)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{(\beta+1)(\beta+2)}|y|^{\beta+2},$
$\displaystyle\varphi_{\beta}^{(2)}(y)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{0}^{\infty}\left(|y-u|^{\beta}+|y+u|^{\beta}\right)\log(1+\operatorname{\rm
e}^{-u})\mathrm{d}u,$ $\displaystyle\varphi_{\beta}^{(3)}(y)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\int_{|y|}^{\infty}|y+u|^{\beta}\log(1+\operatorname{\rm
e}^{-u})\mathrm{d}u,$

and the function $\widehat{g}$ satisfies the identity

$\widehat{g}(f(x))=f(x)^{\kappa}g(x),\quad|x|\text{ sufficiently small}.$

To illustrate the previous result, assume $\kappa=0$, write $y=-Mt^{\alpha}$
and send $t\to\infty$. When $\alpha=0$ the three functions $\varphi_{k}^{(j)}$
are all $\mathcal{O}(1)$. When $0<\alpha<1$, the function $\varphi_{k}^{(3)}$
decays exponentially fast, whereas
$\varphi^{(1)}_{k}=\mathcal{O}(t^{(k+2)\alpha})$ while
$\varphi^{(1)}_{k}=\mathcal{O}(t^{k\alpha})$. In either case, the error term
is $\mathcal{O}(t^{-(1-\alpha)N+3\alpha-2})$, which is in fact decaying only
when $\alpha<(N+2)/(N+3)$. In other words, for a given $\alpha$ the asymptotic
formula (A.4) only becomes truly useful (that is, with a small error) when $N$
is chosen sufficiently large.

###### Proof.

We prove the result for $f(x)=x$, the general case follows along the same
lines of the proof of Proposition A.2.

Fix $\delta>0$ so that $g$ is $C^{\infty}$ on $[0,\delta]$. By replacing $g$
with $g\chi_{[0,a]}$ we may assume that $a=+\infty$. We can then bound

$\left|\int_{\delta}^{\infty}g(u)\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u\right|\leq\|g\|_{L^{1}(0,\infty)}\operatorname{\rm
e}^{-\eta t^{\alpha}-t\delta}=\mathcal{O}(\operatorname{\rm e}^{-\delta
t/2}),$

as we are assuming that $\alpha<1$. Proceeding as in the proof of Proposition
A.1, we obtain

$\int_{0}^{\delta}g(x)\log(1+\operatorname{\rm
e}^{-y-tx})\mathrm{d}x=\sum_{k=0}^{N}\frac{\widetilde{g}^{(k)}(0)}{k!}\int_{0}^{\delta}u^{k-\kappa}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u+R_{N+1}+\mathcal{O}(\operatorname{\rm e}^{-\delta
t/2}),$ (A.5)

with error term satisfying

$|R_{N+1}|\leq\frac{1}{(N+1)!}\|\widetilde{g}^{(k+1)}\|_{L^{\infty}(0,\delta)}\int_{0}^{\delta}u^{N+1-\kappa}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u.$ (A.6)

The integrals appearing in the sum and in the bound for the error above are of
the same form, and now we study them. From (A.3) we see that by taking $t$
large enough we can always assume that the point
$u_{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-y/t$ where the
exponent $y+ut$ changes sign belongs to the interval $(0,\delta)$. We split
the integrals on $(0,\delta)$ into the integrals over $(0,u_{t})$ and
$(u_{t},\delta)$. For the first one, we then express it as

$\displaystyle\int_{0}^{u_{t}}u^{\beta}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u$
$\displaystyle=-\int_{0}^{u_{t}}(y+tu)u^{\beta}\mathrm{d}u+\int_{0}^{u_{t}}u^{\beta}\log(1+\operatorname{\rm
e}^{y+tu})\mathrm{d}u$
$\displaystyle=\frac{1}{(\beta+1)(\beta+2)}\frac{|y|^{\beta+2}}{t^{\beta+1}}+\frac{1}{t^{\beta+1}}\int_{0}^{|y|}|v+y|^{\beta}\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v$
$\displaystyle\begin{multlined}=\frac{1}{t^{\beta+1}}\varphi_{\beta}^{(1)}(y)+\frac{1}{t^{\beta+1}}\int_{0}^{y+t\delta}|y+v|^{\beta}\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v\\\
-\frac{1}{t^{\beta+1}}\int_{|y|}^{y+t\delta}|y+v|^{\beta}\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v,\end{multlined}=\frac{1}{t^{\beta+1}}\varphi_{\beta}^{(1)}(y)+\frac{1}{t^{\beta+1}}\int_{0}^{y+t\delta}|y+v|^{\beta}\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v\\\
-\frac{1}{t^{\beta+1}}\int_{|y|}^{y+t\delta}|y+v|^{\beta}\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v,$

where for the second equality we changed variables $v=-y-tu$. Likewise,

$\int_{u_{t}}^{\delta}u^{\beta}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u=\frac{1}{t^{\beta+1}}\int_{0}^{y+t\delta}|y-v|^{\beta}\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v,$

and combining them

$\int_{0}^{\delta}u^{\beta}\log(1+\operatorname{\rm
e}^{-y-tu})\mathrm{d}u=\frac{1}{t^{\beta+1}}\varphi_{\beta}^{(1)}(y)+\\\
\frac{1}{t^{\beta+1}}\left(\int_{0}^{y+t\delta}(|y-v|^{\beta}+|y+v|^{\beta})\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v-\int_{|y|}^{y+t\delta}|v+y|^{\beta}\log(1+\operatorname{\rm
e}^{-v})\mathrm{d}v\right)$

Using that $y+t\delta\to+\infty$ and the bounds (A.3), it is straightforward
to see that the first integral on the right-hand side is exponentially close
to $\varphi_{k}^{(2)}$ while the second integral is exponentially close to
$\varphi_{k}^{(3)}$. The result then follows using these relations back into
(A.5)–(A.6). ∎

## References

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