Splitting of a gap in the bulk of the spectrum of random matrices

TL;DR

This paper analyzes the probability of two gaps in the eigenvalue spectrum of Gaussian Unitary Ensemble matrices, providing detailed asymptotics for the transition between one large gap and two gaps.

Contribution

It offers explicit uniform asymptotics for gap probabilities and characterizes the transition between single and double gaps in the spectrum.

Findings

Explicit asymptotics for the transition between one and two gaps

Asymptotic terms for the probability of two gaps in the spectrum

Analysis of Toeplitz determinants with symbols on two arcs

Abstract

We consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and two large gaps. For the initial stage of the transition, we explicitly determine all the asymptotic terms (up to the decreasing ones) of the logarithm of the probability. We obtain our results by analyzing double-scaling asymptotics of a Toeplitz determinant whose symbol is supported on two arcs of the unit circle.

Equations

$$K_s(x,y)=\frac{\sin s(x-y)}{\pi (x-y)}.$$

$$P_s(A)=\det (I-K_s{)}_A.$$

$$A=A_1\bigcup A_2,A_1=({\alpha }_1,{\beta }_1),A_2=({\alpha }_2,{\beta }_2).$$

$$\log P_s(A_0)$$

$$c_0$$

$$\frac{d}{ds}\log P_s(A)=s{C}_1+C_2(s)+o(1),$$

$$\frac{d}{ds}\log P_s(A)=-2s{G}_0+\frac{d}{ds}\log \theta (sV;\tau )+\mathcal{O}(s^{-1}).$$

$$\frac{G_1(s)}{s}+\frac{G_2(s)}{s^2}+\cdots +\frac{G_j(s)}{s^j}+\mathcal{O}(s^{-j-1}).$$

$$\theta (z)=\theta (z;\tau )=\sum _{m\in \mathbb{Z}}{e}^{2\pi izm+\pi i\tau m^2}$$

$$r(z)=((z-{\alpha }_1)(z-{\beta }_1)(z-{\alpha }_2)(z-{\beta }_2){)}^{1/2},$$

$${\int }_{A_j}\frac{q(z)dz}{r_{+}(z)}=0,j=1,2,$$

$$\frac{q(z)}{r(z)}=1+\frac{G_0}{z^2}+\mathcal{O}(z^{-3}),$$

$$V=-\frac{1}{\pi }{\int }_{{\beta }_1}^{{\alpha }_2}\frac{q(x)dx}{r(x)},\tau =i\frac{{\int }_{{\beta }_1}^{{\alpha }_2}\frac{dx}{|r(x)|}}{{\int }_{{\alpha }_2}^{{\beta }_2}\frac{dx}{|r_{+}(x)|}}.$$

$$\log P_s(A)=-s^2{G}_0+\log \theta (sV;\tau )+{\hat{G}}_1\log s+c_1+\mathcal{O}(s^{-1}),{\hat{G}}_1=\underset{x\to \infty }{\lim }\frac{1}{x}{\int }_{s_0}^x{G}_1(t)dt.$$

$$D_n(J)=\det {\left(f_{j-k}\right)}_{j,k=0}^{n-1},f_k={\int }_{e^{i\theta }\in J}{e}^{-ik\theta }\frac{d\theta }{2\pi },$$

$$J=J_2=\left\{e^{i\theta }|\theta \in \left(-\pi ,{\theta }_2\right)\cup \left({\theta }_1,\pi \right]\right\}.$$

$$\log D_n(J_2^{(n)})=n^2\log \cos \frac{{\theta }_1-{\theta }_2}{4}-\frac{1}{4}\log \left(n\sin \frac{{\theta }_1-{\theta }_2}{4}\right)+c_0+\mathcal{O}\left(\frac{1}{n\sin \frac{{\theta }_1-{\theta }_2}{4}}\right)$$

$$\underset{n\to \infty }{\lim }{D}_n(J_2^{(n)})=\det (I-K_s{)}_{A_0}$$

$$A=(\alpha ,-\nu )\bigcup (\nu ,\beta ),\alpha <0<\nu <\beta .$$

$$\omega =\frac{s\sqrt{|\alpha \beta |}}{\log (\gamma \nu {)}^{-1}}>0.$$

$$\omega =k+x,k=0,1,2,\dots ,x\in [-1/2,1/2).$$

$$\nu ={\gamma }^{-1}{e}^{-\frac{s\sqrt{|\alpha \beta |}}{k+x}}.$$

$$\log P_s(A)$$

$$+\mathcal{O}(\max \{s{\nu }_0\log {\nu }_0^{-1},1/\log {\nu }_0^{-1},s^{-1}\}),$$

$$c(k)$$

$${\delta }_k(x)$$

$${\kappa }_j=4^{-j-1/2}\sqrt{2j+1}\frac{(2j)!}{j{!}^2},j=1,2,\dots ,{\kappa }_0=1/2,{\kappa }_{-1}=0,$$

$$\log P_s(A)$$

$$+\log \left(1+(\gamma \nu {)}^{1-2|x|}\right)-\frac{1}{4}\log \left(\frac{\beta -\alpha }{2}\sqrt{|\alpha \beta |}\right)+\frac{1}{6}\log 2+6{\zeta }^{\prime }(-1)$$

$$+\mathcal{O}\left(\max \left\{s{\nu }_0\log {\nu }_0^{-1},\frac{1}{\log {\nu }_0^{-1}},\frac{\log {\nu }_1^{-1}}{s}\right\}\right),$$

$$s{\nu }^{1/2+\epsilon }\to \infty ,$$

$$s\nu \log (\gamma \nu {)}^{-1}\to 0$$

$$c(k)=-\sum _{j=0}^{k-1}\log 2\pi {\kappa }_j^2,k=1,2,\dots ,c(0)=0.$$

Full text

1 Introduction

Let $A$ be the union of $m$ open disjoint intervals on $\mathbb{R}$, and $K_{s}$ be the (trace-class) integral operator on $L^{2}(A,dx)$ given by the kernel

[TABLE]

Consider the Fredholm determinant

[TABLE]

For a wide class of random matrix ensembles [1], in particular for the Gaussian Unitary Ensemble, $P_{s}(A)$ is the probability that the set $\frac{s}{\pi}A=\{\frac{s}{\pi}x:x\in A\}$ contains no eigenvalues in the bulk scaling limit where the average distance between the eigenvalues is $1$. In this paper, we are interested in the asymptotics of $P_{s}(A)$ as $s\to\infty$, and we study the transition between a single interval $A_{0}=(\alpha,\beta)$ to the set $A$ composed of 2 disjoint intervals

[TABLE]

Such problems have a rich history, of which we mention some relevant results. For the single interval case $A_{0}=(\alpha,\beta)$,

[TABLE]

as $s\to\infty$, where $\zeta$ is the Riemann zeta-function. The leading term and logarithmic term in (4) were conjectured by des Cloizeaux and Mehta [5] in 1973, while the constant term $c_{0}$ remained undetermined until Dyson [9] conjectured an expression for it in 1976, relying on inverse scattering techniques and the work of Widom [16] on Toeplitz determinants (see below). The constant $c_{0}$ became known as the Widom-Dyson constant. The first rigorous confirmation of the leading term in (4) was given by Widom [17] in 1994. In a landmark paper of 1997, Deift, Its, Zhou [7] were able to confirm the leading term and the logarithmic term, but the proof of the constant $c_{0}$ continued to defy their techniques. Finally, two independent proofs of the constant were later given by Erhardt [10] and the second author [12], and a further third proof given in [8]. The proofs in [12], [8] use Riemann-Hilbert (RH) methods, while [10] uses operator theoretical techniques.

When $A$ is composed of any (fixed) number of intervals, the main term was found and proved by Widom [18] in 1995, where he was also able to identify the next term in the following result:

[TABLE]

as $s\to\infty$. The constant $C_{1}$ is explicitly computable, while $C_{2}(s)$ is an oscillatory function given by a Jacobi inversion problem. In [7], which was mentioned above, the authors were also able to find the full asymptotic expansion for the logarithmic derivative of the determinant on any number of intervals and describe the oscillations in terms of $\theta$-functions. Here we present their result when $A$ is composed of 2 intervals as in (3):

[TABLE]

More precisely, for any $j=1,2,\dots$, the error term here is of the form

[TABLE]

where $G_{j}(s)$, $j=1,\dots$, are bounded periodic functions of $s$. Here

[TABLE]

is the Jacobi Theta-function of the third kind, see e.g. [19]. The constants (in $s$) $V,\tau,G_{0}$, are given in terms of elliptic integrals, and $G_{1}(s),G_{2}(s),\dots$ are given in terms of $\theta$-functions. Let

[TABLE]

with branch cuts on $A$ such that $r(z)>0$ for $z>\beta_{2}$, and let $q(z)$ be the unique monic polynomial of degree $2$ such that

[TABLE]

where $r_{+}(z)$ is the limit of $r(z+i\epsilon)$ as $\epsilon\to 0_{+}$ (where the ”+” side is chosen merely for definiteness). Then $q/r$ has no residue at infinity. Hence as $z\to\infty$, $q/r$ has the form

[TABLE]

which defines the constant $G_{0}$ appearing in (6).

The parameters $V,\tau$ appearing in the arguments of the $\theta$-function in (6) are as follows:

[TABLE]

Integrating (6) in $s$ from some large value $s_{0}$ to $s$, and using the properties of $G_{1}(s)$, Deift, Its, and Zhou concluded that

[TABLE]

The value of the constant $c_{1}$ is unknown as there is no point $s_{0}$ for the lower limit of integration where $P_{s_{0}}(A)$ would be explicitly known.

In this paper we study the transition between the single interval formula (4) and the two-interval formula (13). We obtain an explicit expression (up to the decreasing terms in the expansion of $\log P_{s}(A)$) for the asymptotics in the regime where the length $\ell$ of the interval between the gaps decreases sufficiently fast (slightly faster than $1/s$: see below) as $s\to\infty$. On the other hand we show that the asymptotics of [7], obtained for fixed gaps, can be extended (with proper adjustments) to the regime when $\ell$ is no longer fixed but decreases sufficiently slowly as $s\to\infty$. These two regimes overlap. Thus our analysis provides uniform asymptotics for the whole transition. Note, however, that since the constant in (13) is not determined, the expression for the asymptotics in the second regime is not fully explicit. Obtaining an explicit expression for these asymptotics and establishing the constant in (13) is a separate problem and we plan to address it in future work.

The initial phase of the transition between (4) and (13) resembles the birth of a cut — emergence of an extra interval of support of the limiting eigenvalue density in a unitary ensemble of random matrices: asymptotics for the correlation kernel of the eigenvalues in that case were obtained independently by Bertola & Lee [3], Claeys [4], Mo [14]. From the technical point of view, our analysis is very different as we are dealing with so-called hard edges rather than soft edges in [3, 4, 14] and in the context of a different model, so both the g-function needed in the analysis and the local parametrix are different. Moreover, the works [3, 4, 14] deal with correlation kernels and not determinants.

Consider the Toeplitz determinant whose symbol $f(z)$ is the characteristic function of a subset $J$ of the unit circle $C$:

[TABLE]

where integration is in the positive direction around the unit circle. The proofs of the expansion (4) including the constant term $c_{0}$ in [8, 12] were based on an analysis of the Toeplitz determinant $D_{n}(J_{2})$ where $J_{2}$ is an arc of the unit circle

[TABLE]

The asymptotics of $D_{n}(J_{2})$, as $n\to\infty$, for a fixed arc $J_{2}$ were found by Widom [16]. In [8, 12], Widom’s result was extended to the case of $J_{2}=J_{2}^{(n)}$ varying with $n$ such that $|\theta_{1}-\theta_{2}|\to 0$ sufficiently slowly. Namely,

[TABLE]

as $n\to\infty$, uniformly for $\frac{s_{0}}{n}\leq\frac{\theta_{1}-\theta_{2}}{2}\leq\pi-\epsilon$, for $\epsilon>0$ and with $s_{0}$ sufficiently large. Asymptotics (4) are obtained from (16) by using the fact that

[TABLE]

for fixed $s$ and by taking the limit in (16) as $n\to\infty$ with $\theta_{1}=\frac{2s\beta}{n}$ and $\theta_{2}=\frac{2s\alpha}{n}$, where $\alpha,\beta$ are fixed. The approach of the present paper is based on an analysis of $D_{n}(J)$ where $J=J^{(n)}$ is the union of 2 arcs $J^{(n)}=J_{1}^{(n)}\cup J_{2}^{(n)}$, with $J_{1}^{(n)}\subset C\setminus J_{2}^{(n)}$ of sufficiently small length in comparison with $C\setminus J_{2}^{(n)}$ (see Theorem 1.2 below). We obtain our results on the sine-kernel determinant by taking the limit $n\to\infty$ of $D_{n}(J^{(n)})$. However, we believe Theorem 1.2 below to be of independent interest for a future study of Toeplitz determinants with symbols supported on several arcs.

1.1 Results

The kernel (1) is translationally invariant and so we can assume the following form for $A$:

[TABLE]

In this paper we provide the asymptotics of $\log P_{s}(A)$ (including the constant term) in the double scaling limit as $s\to\infty$ while $\nu\to 0$ in such a way that $s\nu\log\nu^{-1}\to 0$, and connect these asymptotics with those of [7].

Let $\gamma=\frac{1}{8}(\beta^{-1}-\alpha^{-1})$ and

[TABLE]

Clearly, $\omega$ is uniquely represented in the form

[TABLE]

We note that $\nu$ has the form

[TABLE]

We prove the following:

Theorem 1.1**.**

As $s\to\infty$, uniformly for $\nu\in(0,\nu_{0})$, where $s\nu_{0}\log\nu_{0}^{-1}\to 0$,

[TABLE]

*where $G$ is the Barnes G-function, and where $\kappa_{j}$ is the leading coefficient of the Legendre polynomial of degree $j$ orthonormal on the interval $[-2,2]$, given by *

[TABLE]

and $\log P_{s}(A_{0})$ is given in (4).

As $s\to\infty$, uniformly for $\nu\in(\nu_{1},\nu_{0})$, where $s\nu_{0}\log\nu_{0}^{-1}\to 0,\,\,\frac{s}{\log\nu_{1}^{-1}}\to\infty$ (i.e., $k\to\infty$), formula (22) reduces to

[TABLE]

where $\zeta$ is the Riemann zeta-function.

Remark 1.1**.**

Note that if $k=0$ and $s\to\infty$ while $x\in(0,1/2-\epsilon)$ for $\epsilon>0$, then (22) shows that $\frac{P_{s}(A)}{P_{s}(A_{0})}\to 1$.

Remark 1.2**.**

As we show in Section 5 (Lemma 5.1), the Deift-Its-Zhou asymptotics (6) for 2 fixed gaps where we set $\alpha_{2}=-\beta_{1}=\nu$, $\alpha_{1}=\alpha$, $\beta_{2}=\beta$, can be extended (with a worse error term) to the region where $\nu\to 0$ in such a way that

[TABLE]

for any $\varepsilon>0$. Clearly, this region overlaps with the region of validity

[TABLE]

of Theorem 1.1. For example, $\nu=s^{-3/2}$ belongs to both regions. In Remark 5.1 we explicitly show the coincidence of the main (order $s^{2}$) asymptotic terms. Full explicit formulas for this matching will be a subject of future work.

Remark 1.3**.**

The function $c(k)$ can alternatively be described in terms of the coefficients (23) of the Legendre polynomials:

[TABLE]

Formula (25) shows that $\delta_{k}(x)+c(k)$ is continuous also at the points $|x|=1/2$.

Remark 1.4**.**

The rescaled sine process (with expected distance between particles $\pi/s$) is the determinantal point process with the $m$’th correlation function $\rho_{m}$, for $m=1,2,\dots$, given by

[TABLE]

Consider the rescaled sine process conditioned to have no eigenvalues in $A$. Denote this process by $\mathbb{P}_{A}$ and its $m$’th correlation function by $\rho_{m}^{A}$. In Section 4.2, we show that for $x_{1},\dots,x_{m}\in(-1,1)$,

[TABLE]

as $s\to\infty$ and $\nu\to 0$ such that $k\in\mathbb{N}$ and $|x|<1/2$ remain fixed, where

[TABLE]

and $L_{k}$ is the Legendre polynomial of degree $k$, orthonormal on $[-2,2]$:

[TABLE]

Recall that for a set $B\subset\mathbb{R}$ and a point process $\Lambda$ with its $m$-th correlation function denoted $r_{m}$, we have

[TABLE]

The process with kernel $K_{\textrm{Leg}}$ is a $k$-point process. Thus we obtain from (27) and the first equation of (30) that, as $s\to\infty$ and $\nu\to 0$ such that $k\in\mathbb{N}$ and $|x|<1/2$ remain fixed, the expected number of $(k+1)$-tuples of $\mathbb{P}_{A}$ on $\left(-\nu,\nu\right)$ converges to [math], while the expected number of $k$-tuples on the same interval converges to $1$. It follows from the second equation of (30) that

[TABLE]

Thus the asymptotics of $\log P_{s}(A)$ as $s\to\infty$ depends on the value of $\nu$, and we give an overview of the various scaling limits:

• •

If $\nu=0$, the asymptotics are given by (4).

• •

If $\nu\to 0$ as $s\to\infty$, such that $s\nu\log\nu^{-1}\to 0$, the asymptotics are given by Theorem 1.1.

• •

If $\nu\log\nu^{-1}$ is of order $1/s$ or larger, the asymptotics of Theorem 1.1 breaks down and the transition to the asymptotic formula (13) containing $\theta$-functions takes place. This is discussed in Section 5.

• •

If $\nu>0$ is fixed, the asymptotics are given by the $\theta$-function regime (13).

For Toeplitz determinants, we obtain the following result. Let $D_{n}(J)$ be given by (14) with $J=J^{(n)}=J_{1}^{(n)}\cup J_{2}^{(n)}$ where

[TABLE]

with some $\alpha<0<\nu(s)<\beta$. Then, with the notation of Theorem 1.1, we have

Theorem 1.2**.**

*As $s,n\to\infty$, uniformly for $\nu\in(0,\nu_{0})$, where $s\nu_{0}\log\nu_{0}^{-1}\to 0$ and

$s^{3}/n\to 0$,*

[TABLE]

where the expansion of $\log D_{n}\left(J^{(n)}_{2}\right)$ is given in (16) with $\theta_{1}=2s\beta/n$, $\theta_{2}=2s\alpha/n$.

We use Theorem 1.2 to prove Theorem 1.1.

Proof of Theorem 1.1. It is well-known that

[TABLE]

as $n\to\infty$ for fixed $s$, a fact which we also prove in the appendix for the reader’s convenience. Taking the limit $n\to\infty$ in (33), we then obtain (22). To obtain (24), we substitute (4) for $P_{s}(A_{0})$, and note that the standard asymptotics of the Barnes G-function $G(z+1)$ as $z\to\infty$

[TABLE]

imply that as $k\to\infty$,

[TABLE]

Furthermore, we note that $2\pi\kappa_{k}^{2}=1+{\cal O}(k^{-1})$ as $k\to\infty$. $\Box$

1.2 Outline of the proof of Theorem 1.2

It remains to prove Theorem 1.2.

Let $-\pi<\theta_{2}<0<\theta_{0}<\theta_{1}<\pi$ and define $J=J_{1}\cup J_{2}$ where $J_{1},\,J_{2}$ are as in Figure 1:

[TABLE]

We denote the complement of $J$ as $\Sigma=\Sigma_{1}\cup\Sigma_{2}$ where

[TABLE]

It follows from the integral representation for Toeplitz determinants (see (295) in the Appendix) that $D_{j}(J)>0$ for all $j\in\mathbb{N}$. Consider the polynomials $\phi_{j}$ for $j=0,1,2,\dots$ given by

[TABLE]

where the leading coefficient $\chi_{j}$ is given by

[TABLE]

and we set $D_{0}(J)=1$. The polynomials $\phi_{j}$ are orthonormal with weight $1$ on $J$:

[TABLE]

Define a $2\times 2$ matrix $Y(z)=Y_{n}(z)$ in terms of these orthogonal polynomials as follows:

[TABLE]

Then $Y$ is the unique solution of the following Riemann-Hilbert (RH) Problem

• (a)

$Y:\mathbb{C}\setminus J\to\mathbb{C}^{2\times 2}$ is analytic;

• (b)

$Y$ possesses $L^{2}$ boundary values $Y_{+}$ and $Y_{-}$ on the $+$ and $-$ side of $J$, respectively, related by the condition:

[TABLE]

• (c)

$Y(z)=(I+\mathcal{O}(1/z))\begin{pmatrix}z^{n}&0\\ 0&z^{-n}\end{pmatrix}$ as $z\to\infty$.

The fact that orthogonal polynomials satisfy a RH problem was first observed for polynomials orthogonal on the real line by Fokas, Its, Kitaev [11], and extended to polynomials orthogonal on the unit circle by Baik, Deift, Johansson [2]. The RH problem provides an efficient tool, via the Deift-Zhou steepest descent method, for the asymptotic analysis of the polynomials, see e.g. [6].

In Section 2 we express the logarithmic derivative of the Toeplitz determinant $\frac{d}{d\theta_{0}}\log D_{n}(J)$ in terms of the polynomials $\phi_{n}$ and $\phi_{n-1}$. These are, in turn, expressed in terms of $Y_{n}$. In Sections 3 and 4 we analyse the RH problem for $Y_{n}$ as $n\to\infty$ in a double scaling limit where $J$ depends on $n$ such that $\theta_{j}=\frac{s}{n}u_{j}$ for $j=0,1,2$; where $s\to\infty$ such that $s^{3}/n\to 0$; where $u_{0}\to 0$ such that $su_{0}\log u_{0}^{-1}\to 0$, while $u_{1}$ and $u_{2}$ remain fixed. As a result, we obtain the asymptotics of $Y_{n}$. Substituting these into the differential identity for $\frac{d}{d\theta_{0}}\log D_{n}(J)$, and integrating with respect to $\theta_{0}$, we obtain the asymptotics of $D_{n}(J)$, where $u_{1}=2\beta,\,\,u_{2}=2\alpha$, and $u_{0}=u_{0}(\nu)$ is a function of $\nu$, which proves Theorem 1.2.

2 Differential Identity

We will now obtain the following:

Proposition 2.1**.**

(Differential identity) Let $a=e^{i\theta_{0}}$. The Toeplitz determinant $D_{n}(J)$ satisfies

[TABLE]

where

[TABLE]

and $J$ was given in (37).

Proof.

From the definition of the orthogonal polynomials it is clear that

[TABLE]

The orthogonality conditions imply that

[TABLE]

and similarly,

[TABLE]

By (45)–(47) we obtain:

[TABLE]

On the other hand, one can express $F$ given in (44) in terms of the orthogonal polynomials:

[TABLE]

Now the Christoffel-Darboux formula for orthogonal polynomials gives

[TABLE]

(see eg. [12]), and hence (48) can be written as

[TABLE]

Since, by (50) and orthogonality, $\int_{J}F(z)\frac{d\theta}{2\pi}=n$, we obtain

[TABLE]

upon which Proposititon 2.1 follows immediately.∎

3 Analysis of Riemann-Hilbert problem

We start by setting $\theta_{0}=0$ so that $J_{1}$ is a point, and then let $J_{1}$ develop into an arc. Throughout the rest of the paper we use the notation

[TABLE]

We let $s,n\to\infty$ such that $s^{3}/n\to 0$, and let $u_{0}\to 0$ as $s\to\infty$ such that $su_{0}\log u_{0}^{-1}\to 0$, while $u_{2}<0<u_{1}$ remain constant. Denote $\Sigma^{o}=\Sigma_{1}^{o}\bigcup\Sigma_{2}^{o}$ where

[TABLE]

Let $g_{1}$ be the function:

[TABLE]

where the square root has branch cut on $J_{2}$ and is positive as $z\to+\infty$, and the logarithm $\log x$ has a branch cut for $x<0$ and is positive for $x>1$. At infinity,

[TABLE]

The boundary values of the function $g_{1}$ satisfy

[TABLE]

and at $b_{1},b_{2}$ we have

[TABLE]

Alternatively, for $z=e^{i\theta}\in\Sigma^{o}$ we can write $g_{1}$ in the following form:

[TABLE]

On the $+$ and $-$ side of $J_{2}$, we have

[TABLE]

from which it follows that $e^{g_{1}}$ maps the $+$ side of $J_{2}$ to $\Sigma^{o}$ and the $-$ side to $J_{2}$, and that $e^{g_{1}}$ maps $\mathbb{C}\setminus J_{2}$ to the outside of the unit disc.

Set

[TABLE]

for $j=1,2$. Note that $B_{1}$ and $B_{2}$ remain bounded as $s/n\to 0$. Then

[TABLE]

where $H(w)$ is analytic in $w$ at the point [math]. Thus $g_{1}$ has the following expansion in $w$ at the point $w=0$

[TABLE]

where

[TABLE]

as $\frac{s}{n}\to 0$.

Define

[TABLE]

where the square root has a branch cut on $J_{2}$, and is positive as $z\to+\infty$. Let

[TABLE]

where integration is taken in counter-clockwise direction. It is easily verified by differentiation that

[TABLE]

for any constant $\widetilde{C}$ and some function $C(z)$. Thus

[TABLE]

The function $h$ has a logarithmic singularity at $z=1$ and a jump on $\Sigma_{2}^{o}\bigcup J_{2}$, such that

[TABLE]

The jump conditions (69) also imply that

[TABLE]

As $z\to\infty$,

[TABLE]

On the interval $\Sigma^{o}$ we can alternatively write $h$ in the following form:

[TABLE]

With the notation of (61) we can write

[TABLE]

Then we can expand $h$ at the point $z=1$:

[TABLE]

where

[TABLE]

as $\frac{s}{n}\to 0$.

We define the $g$-function by:

[TABLE]

where $\Omega>0$ is a constant yet to be fixed. The jump conditions for $g_{1}$ and $h$ imply that

[TABLE]

We define the local variable $\zeta$ on a disc $U_{0}$ containing the interval $J_{1}$ (but not the points $b_{1},b_{2}$), by

[TABLE]

The jump conditions for $g$ imply that the function $\zeta$ is analytic in $U_{0}$. The precise radius of $U_{0}$ will be determined later on by requiring that the mapping $\zeta$ be conformal on $U_{0}$.

Since $e^{g_{1}}$ maps $\mathbb{C}\setminus J_{2}$ to the exterior of the unit disc, we have

[TABLE]

For $u_{0}<\epsilon$ with some $\epsilon>0$, it follows from (72) that

[TABLE]

Now consider, as a function of $\Omega$,

[TABLE]

By (79) and (80) it follows that if we let $\Omega=+\infty$ in (78) then (81) is smaller than 2, and if we instead set $\Omega=+0$ then (81) is equal to $+\infty$. Since (82) is monotone in $\Omega$, there exists, for $u_{0}<\epsilon$, a unique value for $\Omega>0$ such that

[TABLE]

We define $\Omega$ so that $\zeta$ satisfies (82). From (59) and (72), it follows that $\zeta(\Sigma^{o})\subset\mathbb{R}$. By (63) and (74) we have the following expansion at the point $z=1$:

[TABLE]

In what follows, it will be apparent that $\Omega\to 0$ in the limit $s,n\to\infty$ and $u_{0},s/n\to 0$. Moreover, by (64) and (75),

[TABLE]

Substituting these expansions into (83), which we in turn substitute into (82), we obtain

[TABLE]

or upon taking the logarithm,

[TABLE]

Therefore,

[TABLE]

Using the definition of $g_{1}$, $h$ and $\zeta$ in (55), (68), (78), and the expansion of $\Omega$ in (87), it is easily seen that there are constants $m_{1}<m_{2}$ independent of $s,n,u_{0}$ such that $\zeta^{\prime}(z)$ has at least one zero in the set

[TABLE]

By (84) and expansion (87), $m_{1}$ may be chosen such that

[TABLE]

as $\frac{sw}{n\Omega}\to 0$ and so $\zeta$ is conformal on the following disc

[TABLE]

for $u_{0},s/n<\epsilon$ for some fixed $\epsilon>0$. Thus we define $U_{0}$ to be the set (90).

We define $\widetilde{g}$ as

[TABLE]

and $T$ as

[TABLE]

It follows by (77) that $T$ satisfies the following RH problem:

• (a)

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

• (b)

$T$ has the following jumps on $C$:

[TABLE]

• (c)

As $z\to\infty$,

[TABLE]

The jump of $T$ on $J_{2}$ factorizes as

[TABLE]

Define

[TABLE]

Then the jumps of $e^{\phi}$ are induced by $g$ and we obtain by (77) that for $z\in J_{2}.$

[TABLE]

and similarly

[TABLE]

We proceed to open the lenses around $J_{2}$ as in Figure 2.

Proposition 3.1**.**

For $z$ on the edges of the lense $\Gamma_{S}^{\textrm{in}}\cup\Gamma_{S}^{\textrm{out}}$ in Figure 2 such that $|z-b_{1}|,|z-b_{2}|>\epsilon s/n$ for some fixed $\epsilon>0$, there exists a constant $C>0$ independant of $s,n,z$ such that

[TABLE]

as $s,n\to\infty$ and for $u_{0}$ sufficiently small.

Proof.

Since $e^{g_{1}}$ sends $\mathbb{C}\setminus J_{2}$ to the outside of the unit disc, it is clear that for $|z|<1$ we have

[TABLE]

Let $g_{1}^{(-)}$ denote the function defined as in (55), but with $+((z-b_{1})(z-b_{2}))^{1/2}$ replaced with $-((z-b_{1})(z-b_{2}))^{1/2}$ Then $e^{g_{1}^{(-)}}$ maps $\mathbb{C}\setminus J_{2}$ to the inside of the circle and for $z\in\mathbb{C}\setminus J_{2}$ we have the relation

[TABLE]

It follows that if $|z|>1$, then

[TABLE]

Using (98), (100), the definition of $g$, and the fact that $\Omega=\mathcal{O}(s/(n\log u_{0}^{-1})),$ as $s/(n\log u_{0}^{-1})\to 0$, it follows that $e^{-\phi(z)}$ lies in interior of the unit disc for $z$ sufficiently close to the interval $J_{2}$, and in particular that

[TABLE]

uniformly for $z$ on the lense that is opened around $J_{2}$ in Figure 2, except near the endpoints $b_{1}$ and $b_{2}$, for some constant $c>0$.

Consider $h(z)$ and $g_{1}(z)$ at $z=b_{1}$. Let $w_{1}=\frac{n}{s}(z-b_{1})$. From (68) we have, with $B_{j}=\frac{n}{s}(1-b_{j})$,

[TABLE]

It follows from (69) and (102) that $h(z)/w_{1}^{1/2}$ is analytic at $w_{1}=0$, and we have

[TABLE]

Likewise, we let $w_{2}=\frac{n}{s}(z-b_{2})$. Then at the point $w_{2}=0$ we have the expansion

[TABLE]

where $\pm$ means $+$ on the $+$side and $-$ on the $-$side of the unit circle $C$ (so the jumps agree with (69)).

We evaluate $g_{1}$ using the definition (55):

[TABLE]

From (105) and (57), we have that $(g_{1}(z)-\frac{\log z}{2})/w_{1}^{1/2}$ is analytic at $w_{1}=0$, and that at the point $w_{1}=0$ we have the expansion

[TABLE]

Likewise we expand $g_{1}$ at the point $w_{2}=0$:

[TABLE]

Consider now a neighbourhood of $b_{1}$. The error terms in (103), (106) are uniform for $0<s/n<\delta$ for some sufficiently small $\delta$. By (103), (106) and the fact that $\Omega=\mathcal{O}\left(\frac{s}{n\log u_{0}^{-1}}\right)$, it follows that there is a constant $C_{1}>0$ independant of $s,n,z$ for $u_{0}$ sufficiently small such that

[TABLE]

as $z\to b_{1},s/n\to 0$. Thus from (108) it follows that there exists $\epsilon,C>0$ such that for all $|z-b_{1}|>\epsilon s/n$ we have

[TABLE]

as $n,s\to\infty$, and for $u_{0}$ sufficiently small. The same may be shown at the point $b_{2}$, concluding the proof. ∎

Let

[TABLE]

Then $S$ satisfies the following RH problem:

• (a)

$S:\mathbb{C}\setminus\Gamma_{S}\to\mathbb{C}^{2\times 2}$ is analytic, where $\Gamma_{S}=C\cup\Gamma_{S}^{in}\cup\Gamma_{S}^{out}$ as shown in Figure 2.

• (b)

On $\Gamma_{S}\setminus\{a,\overline{a},b_{1},b_{2}\}$, $S$ has the following jumps:

[TABLE]

• (c)

As $z\to\infty$,

[TABLE]

3.1 Main parametrix

In the region $\mathbb{C}\setminus(U_{0}\cup U_{1}\cup U_{2})$, we approximate the RH problem for $S$ by a main parametrix $M$, which satisfies the RH problem:

• (a)

$M:\mathbb{C}\setminus\overline{\{J_{2}\bigcup\Sigma_{2}^{o}\}}\to\mathbb{C}^{2\times 2}$ is analytic.

• (b)

On $J_{2}$ and $\Sigma_{2}^{o}$, $M$ has the following jumps:

[TABLE]

• (c)

As $z\to\infty$,

[TABLE]

A solution to the RH problem for $M$ is given by

[TABLE]

where $F$ is a constant matrix and

[TABLE]

with branch cuts on $J_{2}$ and such that $\gamma_{1}(z)\to 1$ and $\gamma_{2}(z)\to 0$ as $z\to\infty$. For $y\in\mathbb{R}$, let $\left<y\right>$ be defined such that

[TABLE]

Then $D$ is given by

[TABLE]

where it follows from the jumps of $h$ (69) that $D$ is analytic for $z\in\mathbb{C}\setminus(\overline{\Sigma_{2}^{o}\bigcup J_{2}})$, and

[TABLE]

The function $M$ defined in (112) will solve the RH problem for $M$ with any constant matrix $F$, which we will define later in (128). The reason for the prefactor $I+\frac{F}{z-1}$ in (112), which does not affect the jump conditions for $M$, will become apparent later on.

3.2 Model RH problem $\Phi$

Consider the following RH problem for $\Phi(\zeta;k)$, where $k\in\mathbb{N}$:

• (a)

$\Phi(\zeta):\mathbb{C}\setminus[\eta_{1},\eta_{2}]\to\mathbb{C}^{2\times 2}$ is analytic for given $\eta_{1}<\eta_{2}$.

• (b)

$\Phi$ has $L^{2}$ boundary values for $\zeta\in(\eta_{1},\eta_{2})$ satisfying

[TABLE]

• (c)

As $\zeta\to\infty$,

[TABLE]

It is well-known by the standard theory and is easy to verify that the unique solution to this RH problem is given by

[TABLE]

where $L_{k}$ are the Legendre polynomials of degree $k$ with positive leading coefficients, orthonormal on $(\eta_{1},\eta_{2})$:

[TABLE]

and we denote the first 3 leading coefficients as follows:

[TABLE]

Writing the large $\zeta$ expansion of (118) and using orthogonality in the second column, we obtain that

[TABLE]

When $k=0$ we have

[TABLE]

It is well known that $L_{k}$ has the explicit representation for $k\geq 0$:

[TABLE]

where $\binom{0}{0}=1$. As a consequence, the coefficients in (120) are given by

[TABLE]

for $k\geq 0,1,2$ respectively. From (118), (123), (124), it follows that for $k\geq 1$,

[TABLE]

3.3 Local parametrix at $1$

Recall that $U_{0}$ defined by (90) is an open disc containing $J_{1}$ and that as $s,n\to\infty,u_{0}\to 0$ the radius of $U_{0}$ is of length $\epsilon s/(n\log u_{0}^{-1})$ for some $\epsilon>0$. On $U_{0}$ we defined a local variable $\zeta$ in (78). We define the local parametrix $P$ on $U_{0}$ by

[TABLE]

where $\Phi$ is given by (118), and where $E$ is an analytic function on $U_{0}$ given by

[TABLE]

with constant matrices $F$ and $X$ defined below. From (59) and (72) we see that $\zeta(J_{1})\subset\mathbb{R}$. We let $\eta_{1}=\zeta(\overline{a})$ and $\eta_{2}=\zeta(a)$. Then $\zeta(J_{1})=(\eta_{1},\eta_{2})$, and so $P(z)$ has a jump on $J_{1}$ induced by that of $\Phi$ on $(\eta_{1},\eta_{2})$. $F$ is a constant, nilpotent matrix

[TABLE]

where

[TABLE]

where $h(\infty)$ was defined in (71) and $X$ is a constant matrix given in terms of elements of (121)

[TABLE]

The factor $I-X/\zeta(z)$ in $E(z)$ is needed to cancel the would be $u_{0}^{1-2|x|}\log u_{0}^{-1}$ non-smallness in the matrix elements of $\Delta_{1}^{(1)}$ originating from $\Phi_{1,12}B_{11}^{2}$ for $0\leq x<1/2$ and $\Phi_{1,21}B_{11}^{-2}$ for $-1/2\leq x<0$ (see Proposition 3.2 below) so that $P$ and $M$ match to the main order on the boundary $\partial U_{0}$ for all $x\in[-1/2,1/2)$. This factor, however, has a pole at $z=1$, but we need $E(z)$ to be analytic in $U_{0}$. As is easy to verify, the analyticity of $E(z)$ (i.e. the absence of a pole at $1$) is achieved by choosing $F$ as defined in (128).

Proposition 3.2**.**

As $u_{0}\to 0$ and $s,n\to\infty$, we have the following matching condition uniformly on the boundary $\partial U_{0}$:

[TABLE]

where

[TABLE]

and $\Delta_{1}^{(1)}$, $\Delta_{2}^{(1)}$ and $\Xi^{(1)}$ are given by (138) below. We have, uniformly for $z$ on the boundary $\partial U$,

[TABLE]

Proof.

First, assume that $k$ is bounded. Since

[TABLE]

we have on the boundary $\partial U$ that (recall (76), (112), (117), (127))

[TABLE]

It follows from (64), (87), (129) that

[TABLE]

as $u_{0},s/n\to 0$ and $s,n\to\infty$. Denote

[TABLE]

From (135), the boundedness of $\widetilde{h},\widetilde{h}^{-1},\psi,\psi^{-1},\gamma_{1}(z),\gamma_{2}(z)$ for $z\in\partial U_{0}$, and the fact that the radius of $U_{0}$ equals $\epsilon\frac{s}{n\log u_{0}^{-1}}$ for some $\epsilon>0$, we have

[TABLE]

as $u_{0}\to 0$ and $s,n\to\infty$, uniformly for $z\in\partial U_{0}$. From (134) and (136) it follows that $\Delta_{1}^{(1)},\Delta_{2}^{(1)}$ take the form

[TABLE]

From (63)–(64), and recalling (87) and the fact that $w=\mathcal{O}\left(\frac{1}{\log u_{0}^{-1}}\right)$ as $u_{0}\to 0$, it follows that

[TABLE]

as $u_{0}\to 0$, uniformly on the boundary $\partial U_{0}$. Similarly, substituting (84) into (83) and recalling (87), we have

[TABLE]

as $u_{0}\to 0$, uniformly on the boundary $\partial U_{0}$. From (83) and (84) we have

[TABLE]

as $u_{0}\to 0$. Using (141) it follows from (121) and (124) that

[TABLE]

as $u_{0}\to 0$ (for finite $k$). Combining (137)–(142) the proposition is proven for bounded $k$.

Now consider $k\to\infty$. From Stirling’s formula we have

[TABLE]

as $k\to\infty$. Thus (135) holds uniformly for $k\in\mathbb{N}$. We study the particular double scaling limit where $k,\zeta\to\infty$, and from (85), (141) we have that $\eta_{1}+\eta_{2}\to 0$ in such a manner that $k/\zeta,k(\eta_{1}+\eta_{2})=\mathcal{O}(u_{0}s)\to 0$. Thus using (143) we find that as $k\to\infty$

[TABLE]

We also find that as $k\to\infty$ and $(\eta_{1}+\eta_{2})\to 0$ such that $(\eta_{1}+\eta_{2})k\to 0$

[TABLE]

Thus we know the large $k,\zeta$ behaviour of $\zeta^{-1}\Phi_{1},\,\zeta^{-2}\Phi_{2}$, and upon substituting into (138) this yields (132). It remains to calculate the error terms of order $\zeta^{-3}\Phi_{3}$ and higher, and in particular establish their behaviour as $k\to\infty$ with $\zeta$. We rely here on the work by Kuijlaars, McLaughlin, Van Assche and Vanlessen in [13] where the authors found uniform error terms for the Legendre polynomials $L_{k}$ as $k\to\infty$. In the remaining part of the proof of the proposition, we let $\widehat{Y},\widehat{R},\widehat{N}$ denote the functions $Y,R,N$ found in [13]. We compare $\widehat{Y}$ to $\Phi$ from (118) in the present paper:

[TABLE]

where the parameter $n$ in [13] is set to be $k$ here. For $y$ bounded away from $[-1,1]$ it follows from equations (3.1), (4.2), (5.5), (7.1) in [13] that

[TABLE]

By the form of $\widehat{N}$ in (147) above and formula (8.11) in [13] it is clear that

[TABLE]

as $\zeta\to\infty$, where $\chi_{1}$ and $\chi_{2}$ are bounded for $k\in\mathbb{N}$ and the $\mathcal{O}(\zeta^{-3})$ term is uniform for $k\in\mathbb{N}$. As $\zeta,k\to\infty$ and $\eta_{1}+\eta_{2}\to 0$ such that $k/\zeta\to 0$, $k(\eta_{1}+\eta_{2})\to 0$, we have

[TABLE]

It follows from (147)-(148) that as $\zeta,k\to\infty$ and $\eta_{1}+\eta_{2}\to 0$

such that $k/\zeta\to 0$, $k(\eta_{1}+\eta_{2})\to 0$,

[TABLE]

where, in particular, $\chi_{1}$ and $\chi_{2}$ are bounded for $k\in\mathbb{N}$ and the $\mathcal{O}(\zeta^{-3})$ term is uniform for $k\in\mathbb{N}$. By comparing (149) with (117) it follows that as $\zeta,k\to\infty$ and $\eta_{1}+\eta_{2}\to 0$ such that $k/\zeta\to 0$, $k(\eta_{1}+\eta_{2})\to 0$,

[TABLE]

∎

3.4 Model RH problem $\Psi$

The following RH problem has a solution in terms of Bessel functions.

• (a)

$\Psi:\mathbb{C}\setminus\overline{\Gamma_{\Psi}}\to\mathbb{C}^{2\times 2}$ is analytic, where $\Gamma_{\Psi}=\mathbb{R}^{-}\cup\Gamma^{\pm}_{\Psi}$, with $\Gamma^{\pm}_{\Psi}=\{xe^{\pm\frac{2\pi}{3}i}:x\in\mathbb{R}^{+}\}$, and with orientation taken in the direction of increasing real part.

• (b)

$\Psi$ has continuous boundary values $\Psi_{+},\Psi_{-}$ on $\Gamma_{\Psi}$ satisfying the following jump conditions:

[TABLE]

• (c)

As $\zeta\to\infty$, $\Psi$ has the following asymptotics:

[TABLE]

• (d)

As $\zeta\to 0$, the behaviour of $\Psi$ is

[TABLE]

This RH problem has a solution given in [13], in terms of Bessel functions. For definitions and properties of Bessel functions see [15]. We take the principal branches of the Bessel functions. For $|\arg\zeta|<2\pi/3$, we have

[TABLE]

For $2\pi/3<\arg\zeta<\pi$ we the solution is given by

[TABLE]

For $-\pi<\arg\zeta<-2\pi/3$ it is defined as

[TABLE]

We have the following useful asymptotics as $\zeta\to 0$ for $I_{0}$ and $K_{0}$:

[TABLE]

3.5 Local parametrix at $b_{1}$ and $b_{2}$

Let $U_{1}$ and $U_{2}$ be discs of radius $\frac{\epsilon s}{n}$ for some fixed but sufficiently small $\epsilon>0$, centered at $b_{1}$ and $b_{2}$ respectively. Recalling $w_{j}=\frac{n}{s}(z-b_{j})$ for $j=1,2$, we have $|w_{j}|=\epsilon$ on $\partial U_{j}$. For $z\in U_{1}$, define

[TABLE]

where $\phi$ was defined in (94). Recall the notation $B_{j}=\frac{n}{s}(1-b_{j})$ for $j=1,2$. By (103) and (106) we have the following expansion of $\zeta_{1}(z)$ for $w_{1}$ in a neighbourhood of [math]:

[TABLE]

and by considering (77) in addition, one verifies that $\zeta_{1}$ is analytic on $U_{1}$.

Recall from (58), (70) that $\phi_{\pm}(b_{2})=\pm\Omega i$ and define

[TABLE]

where $\mathcal{D}$ denotes the unit disc. Then $\widetilde{\phi}:U\setminus J_{2}\to\mathbb{C}$ is analytic, with a square root singularity at $b_{2}$. We define the local variable

[TABLE]

which is analytic on $U_{2}$. Then, by (104) and (107), $\zeta_{2}(z)$ has the following expansion at $w_{2}=0$:

[TABLE]

and by considering (77) in addition, one verifies that $\zeta_{2}$ is analytic on $U_{2}$.

The local parametrix is given by

[TABLE]

on the $\pm$ side of the contour $C$, where $\phi_{+}(b_{1})=0$ and $\phi_{+}(b_{2})=\Omega i$. As a consequence of the expansions of $\zeta_{j}$ above, we have

[TABLE]

and recalling the definition of $M$ in (112), one may verify that $E_{j}$ is analytic on $U_{j}$. Recalling the jumps of $\phi$ in (95)–(96) and jumps of $g$ in (77), one verifies that the jumps of $P_{j}$ match those of $S$ on $U_{j}$.

Since, recalling (129), $F=\mathcal{O}(s/n)$ as $s/n\to 0$ while $D(\infty)$ remains bounded, we have that $E_{j}$ is uniformly bounded on $\partial U_{j}$, and it follows that uniformly for $z\in\partial U_{j}$ we have the following matching condition of $M$ and $P_{j}$

[TABLE]

as $s\to\infty$. A simple calculation yields

[TABLE]

as $z\to b_{1}$, where the $\mathcal{O}(1)$ part is analytic on $U_{1}$. Similarly, as $z\to b_{2}$ we have:

[TABLE]

again with $\mathcal{O}(1)$ analytic.

3.6 Small norm RH problem

We define $R$ as follows:

[TABLE]

Using standard small norm analysis, it follows from Proposition 3.2, (168)-(169) and the fact that the contour lengths are $\partial U_{j}=\mathcal{O}(s/n)$ for $j=1,2$ as $s/n\to 0$ and $\partial U_{0}=\mathcal{O}(s/n(\log u_{0}^{-1}))$ as $s/n,u_{0}\to 0$ that given $\epsilon>0$,

[TABLE]

uniformly for $|z-1|>\epsilon$.

If $z\in U_{0}$ then it follows from Proposition 3.2, and (168)-(169) that

[TABLE]

where $||R_{j}||$ is the largest element of $R_{j}$ in absolute value for $j=1,2$, and where the matrices $R_{j}$ are given by

[TABLE]

with clockwise orientation taken in the integrals.

4 Asymptotic analysis of the differential identity and correlation functions

4.1 Asymptotics of $\chi_{n}$

From (42) we have

[TABLE]

By the transformations $Y=\widetilde{g}^{-n\sigma_{3}}Te^{ng\sigma_{3}}$ and $T=S=RM$ at $z=0$, (see (92), (110), (170)) and recalling that $\chi_{n}$ is positive, we find from (174) that

[TABLE]

From the definition of $g$ in (76) and $\widetilde{g}$ in (91) it follows by computing $g_{1}(0)$, $h(0)$ in (55), (68) that

[TABLE]

so that

[TABLE]

By (171)

[TABLE]

as $n\to\infty$. Furthermore, we note that $F=\mathcal{O}(s/n)$ and that $\gamma_{2}(0)=-1+\mathcal{O}(s/n)$ as $s/n\to 0$, and substitute this into the definition of $M$ in (112) to find

[TABLE]

as $s/n\to 0$. Substituting (178) into (177) and recalling the notation (129) it follows that

[TABLE]

as $s/n\to 0$. We note that $\widetilde{g}=1+\mathcal{O}(s/n(\log u_{0}^{-1}))$ as $s/n(\log u_{0}^{-1})\to 0$, and thus we have

[TABLE]

as $s/n\to 0$.

4.2 Convergence of correlation functions

Let $H_{n}(x,y)$ be the kernel built out of the orthogonal polynomials on $J$

[TABLE]

By the Christoffel-Darboux formula, $H_{n}$ also has the following useful form

[TABLE]

where $z_{j}=\exp\left(\frac{2sy_{j}i}{n}\right)$ for $j=1,2$. Let $\widehat{K}_{n}$ be defined similarly, but for the special case where $J=C$, namely:

[TABLE]

Let $\rho^{(n)}_{m}$ be the $m$’th correlation function of the determinantal point process with correlation kernel $\widehat{K}_{n}$, and let $\rho^{(n,A)}_{m}$ be the $m$’th correlation function of the same process conditioned to have no points in $A=(\alpha,-\nu)\cup(\nu,\beta)$. Then

[TABLE]

The two correlation functions are also related as follows:

[TABLE]

Similarly, we can write $\rho_{m}^{A}$ in terms of $\rho_{m}$ (both defined in Remark 1.4):

[TABLE]

The infinite sums (185) and (186) can be seen to converge for fixed $s$ by Hadamard’s inequality. Since

[TABLE]

as $n\to\infty$, it follows by formulas (186) and (185) that

[TABLE]

as $n\to\infty$ for fixed $s$ (similarly to convergence of the determinants (34), see the Appendix).

By the definition of $Y$ in (42) and the formula for $H_{n}$ in (182) we have

[TABLE]

where $z_{j}=\exp\left(\frac{2sy_{j}i}{n}\right)$ for $j=1,2$. For the asymptotics of the correlation kernel we are less ambitious and choose not to proceed with all the detail in last section, and work with $|x|$ bounded away from $1/2$ as $n\to\infty$. Since the intention of $F$ and $X$ was to obtain uniform asymptotics up to the points $|x|=1/2$, we can let $F,X=0$ in $M$ and $P$ when we consider $|x|$ bounded away from $1/2$. Then, in place of Proposition 3.2, we have $PM^{-1}=I+o(1)$ as $u_{0}\to 0$ and $s,n\to\infty$ such that $s/n\to 0$ and $k\in\mathbb{N},|x|<1/2$ remain fixed, uniformly on the boundary $\partial U_{0}$. Thus $R=I+o(1)$ in the same limit, and tracing back the transformations $Y\to T=S=RP$ we have, by (92), (110), (170), that for $z\in J_{1}$:

[TABLE]

where $\widetilde{g}$ is given in (91), and $\widetilde{h}$ in (129). Thus it follows by (118), (180), the fact that $\widehat{g_{1}}$ is real to the main order and that

[TABLE]

as $s/n\to 0$, that as $u_{0}\to 0$ and $s,n\to\infty$ such that $s/n\to 0$, while $k\in\mathbb{N}$ and $|x|<1/2$ remain fixed we have

[TABLE]

Since $\eta_{1}=-2+\mathcal{O}(u_{0})$ and $\eta_{2}=2+\mathcal{O}(u_{0})$ as $u_{0}\to 0$, it follows by continuity of the polynomials that $L_{k}^{(\eta_{1},\eta_{2})}$ can be replaced by $L_{k}^{(-2,2)}$ in (192) without modifying the error term. Similarly, by (85), $2|\zeta_{0}|$ can be replaced by $4u_{0}^{-1}$ without modifying the error terms. Thus, combining (192) and (188), we prove the statement in Remark 1.4.

4.3 Expansion of Differential Identity

In this section we start by writing the differential identity in a more convenient form, and find an expansion for it as $s,n\to\infty$ and $u_{0}\to 0$ such that $su_{0}\to 0$ and $s/n\to 0$, before proceeding to integrate it in Section 4.4. Throughout the rest of the paper, the implicit constants in $\mathcal{O}(\dots)$ are independent of $s,u_{0},n$. For example, if we write $\mathcal{O}(\frac{u_{0}}{s}+u_{0}^{2}),$ then in particular it is uniform in $n$, and if we write $\mathcal{O}(1)$, it means this expression is bounded in the double scaling limit described above.

Write the parametrix $P$ in (126) in $U_{0}$ by grouping the factors as follows

[TABLE]

where $A(z)$ and $C(z)$ are by

[TABLE]

By the transformations $Y=\widetilde{g}^{-n\sigma_{3}}Te^{ng\sigma_{3}}$ and $T=S=RM$ at $z=1$, (see (92), (110), (170)) we have for $z\in U_{0}$

[TABLE]

where

[TABLE]

and

[TABLE]

The expression for $C$ in (197) is valid for $k\geq 1$, while for $k=0$, we have

[TABLE]

It follows that

[TABLE]

where we suppress dependency on the variable $z$ on the right hand side, and where ′ denotes differentiation with respect to $z$. Substituting (180), (195), (199) into (43) we find that

[TABLE]

for $z\in U_{0}$. We would now like to evaluate $F(a)$ and $F(\overline{a})$. Since $\zeta\left(e^{i\theta}\right)$ is real for real $\theta$ on $U_{0}$, it follows that $\frac{d}{d\theta}\zeta\left(e^{i\theta}\right)$ is real. Consider the entries of $C$ and recall that $\Phi_{11}(x)$ is real for $x\in\mathbb{R}$, and that $\Phi_{21}(x)$ and $\Phi_{1,12}$ are purely imaginary. By (197), $C_{11}(e^{i\theta})$ is real and so $z\frac{d}{dz}C_{11}(e^{i\theta})$ is purely imaginary, while $C_{21}(e^{i\theta})$ is purely imaginary and $z\frac{d}{dz}C_{21}(e^{i\theta})$ is real. From (59), we recall that $B_{jj}\left(e^{i\theta}\right)$ is real for $j=1,2$. Thus $\overline{B_{11}}B_{22}=\overline{B_{22}}B_{11}=1$. From these observations, we find that

[TABLE]

When $k=0$, both expressions in (201) are equal to [math].

4.3.1 Evaluation of (201)

Using the expansion for $\zeta$ from (83)–(84), and the fact that $\zeta(a)-\zeta(\overline{a})=4$ from (82), we find that

[TABLE]

We substitute the expansion of $\zeta$ from (83) into (197), and recall (121)-(125), to find

[TABLE]

Using the expression

[TABLE]

which follows from (125), we compute the following:

[TABLE]

We also have

[TABLE]

where the derivative is taken with respect to $z$. We now evaluate $RA$. Let $K$ denote the constant

[TABLE]

We have the derivatives of $\gamma_{1}(e^{i\theta})$ and $\gamma_{2}(e^{i\theta})$ with respect to $\theta$ evaluated at $\theta=0$:

[TABLE]

Let $x_{1}$ and $x_{2}$ denote the following functions:

[TABLE]

Then, using (196), (208) and (209), expand $A$. When $0\leq x<1/2$

[TABLE]

where we denote $\gamma_{j}=\gamma_{j}(1)$ for $j=1,2$. When $-1/2\leq x<0$

[TABLE]

We note that

[TABLE]

Recalling (129), (113), (207), it is readily checked that

[TABLE]

and that as $n,s\to\infty,s/n\to 0$,

[TABLE]

From (210), (211), (213)–(214) it follows that

[TABLE]

Thus, from (201) and (205),

[TABLE]

where $r_{1}$ was defined in (203).

Proposition 4.1**.**

We have

[TABLE]

and $|\widetilde{h}|^{-1}=\mathcal{O}(1)$, where $\widetilde{h}$ was given in (129) and $\widetilde{e}$ was defined in (131).

The main term of Proposition 4.1 is easy to calculate from (209) and (113), but we defer the rest of the proof to Section 4.5.

From (125), (203) we obtain that

[TABLE]

Recall that $k={\cal O}(s/\log u_{0}^{-1})$. Combining (201), (206), (215), (218), and using Proposition 4.1 gives us

[TABLE]

4.3.2 Evaluation of (200)

Suppressing $z$ dependence, we write

[TABLE]

Recall that $K=\mathcal{O}(n/s)$, and that $\theta_{0}=u_{0}\frac{s}{n}$. From (210) and (211) we obtain that for $k\geq 1$

[TABLE]

When $k=0$, we have $F_{2},F_{3},F_{4}=0$, while $F_{1}$ is as in (221) for $0<x<1/2$.

From (87) we have that $\Omega={\cal O}\left(\frac{s}{n\log u_{0}^{-1}}\right)$, and thus substituting $k+x=\frac{n\Omega}{2\pi}$ into (85) we have

[TABLE]

Recalling the expansion of $g_{1}$ in (63)–(64) and the definition of $B$ in (127), and substituting the values of $\kappa_{j}$ and $\Phi$ from (124)–(125) into the expansion of $C$ from (203), we obtain that for $k\geq 1$,

[TABLE]

When $k=0$ (and $0<x<1/2$) we have

[TABLE]

Substituting (214), (217), (223) into (221) we find that

[TABLE]

when evaluated at $e^{\pm i\theta_{0}}$. When $k=0$ (and $0<x<1/2$), we have $F_{2},F_{3},F_{4}=0$, while $F_{1}$ is given by

[TABLE]

Finally we find the order of the term which includes $R^{\prime}$ in (200). Using the equation for $R_{1}$ in (173), it is readily seen that

[TABLE]

We recall that the radius of $U_{0}$ is of size $\mathcal{O}\left(\frac{s}{n\log u_{0}^{-1}}\right)$, and that the radius of $U_{j}$ is of size $\mathcal{O}(s/n)$ for $j=1,2$. Thus

[TABLE]

Substituting the asymptotics for $\Delta_{1}^{(1)}$ from (132) and $\Delta_{1}^{(b_{1})},\,\Delta_{1}^{(b_{2})}$ from (167) into (227), it follows that

[TABLE]

Thus, we have from (209), since $\widetilde{h},\,\,\widetilde{h}^{-1},\,\gamma_{j}(1)={\cal O}(1)$, that also

[TABLE]

The formula for $R^{\prime}A$ is given by (210), (211) but with $x_{1},x_{2}$ replaced by the derivatives $x_{1}^{\prime},x_{2}^{\prime}$. Recall that

[TABLE]

From (229) we obtain that

[TABLE]

By substituting (216), (219), (220), (231) into the definition of $F(z)$ from (200), and substituting the resulting expression into the expression for the differential identity (43), we obtain the following proposition.

Proposition 4.2**.**

We have the following asymptotics for $\log D_{n}(J)$, as $u_{0}\to 0$ and $s,n\to\infty$ such that $su_{0}\log u_{0}^{-1}\to 0$ and $s/n\to 0$:

[TABLE]

where the integration variable $\theta=\frac{s}{n}u$, and where the asymptotics of $F_{0}(z)$ are given in (216) and the asymptotics of $F_{1},\,F_{2},\,F_{3},\,F_{4}$ are given in (225).

4.4 Integration of Differential Identity

We evaluate the integral in formula (232) asymptotically to prove Theorem 1.2.

Using (222), (129) we find that

[TABLE]

Letting $k$ in the expression for $\theta_{0}$ in (233) be fixed, we integrate in $\theta_{0}$, denoting

[TABLE]

Note that by (124) with $\eta_{2}-\eta_{1}=4$,

[TABLE]

We integrate $F_{1}$ from (225), changing the variable of integration using (233) and recalling $K$ from (207) and $\rho$ from (233) to find that for $k\geq 1$,

[TABLE]

where $r_{2}$ is given in (223), $u_{0}=u_{0}(k,x)$.

When $k=0$ we have for $x\in[0,1/2)$

[TABLE]

where, in the ${\cal O}$ error term, $u_{0}=u_{0}(k=0,x)$. Similarly, we integrate $F_{4}$ for $k\geq 1$:

[TABLE]

When $k=0$ and $x\in[0,1/2)$, we have $F_{4}=0$. Thus, for $k\geq 1$,

[TABLE]

If $-1/2<x<1/2$, then for $k\geq 1$

[TABLE]

We keep the term $r_{3}$ in (240) as it is not uniform in $x$, and is not small as $x$ approaches $\pm 1/2$.

[TABLE]

Similarly but simpler, using (225) and (233), we find that for $k\geq 1$,

[TABLE]

where $r_{4}=o(1)$. When $|x|<1/2$ we again keep track of the error term

[TABLE]

When $k=0$, we have $F_{2},F_{3},F_{4}=0$, and thus the integral of $F_{1}+F_{2}+F_{3}+F_{4}$ over $[x=0,x=x_{0}<1/2]$ for $k=0$ is given by (237). For any $k\geq 1$ and $-1/2\leq x<1/2$, combining (237), (239), (240), (242)

[TABLE]

By (241) and (243), it follows that for $|x|<1/2$, $\delta_{k}(x)$ is given explicitly as

[TABLE]

Furthermore, by (216) (which holds for all $k\geq 0$) and Proposition 4.1, we find that

[TABLE]

for $k\geq 1$ and $F_{0}=0$ for $k=0$.

4.4.1 Proof of Theorem 1.2

We sum together (244), (246), and substitute the result into (232), to find that

[TABLE]

where $\sum_{j=1}^{0}=0$. The first sum can be evaluated by noting that

[TABLE]

and, it follows that

[TABLE]

The sum with the leading coefficients $\kappa_{j}$ is given by

[TABLE]

where $G$ is the Barnes G-function. By substituting (249), (250) into (247) we find that

[TABLE]

Now define $\alpha=u_{2}/2$, $\beta=u_{1}/2$, and

[TABLE]

Then, by (233),

[TABLE]

It is an easy exercise using (253), the continuity of $w,x$ as functions of $u_{0}$, and (251) to show that

[TABLE]

Substituting the asymptotics from (251) into (254), and using uniformity of the error terms, we prove Theorem 1.2.

4.5 Proof of Proposition 4.1

We consider the small norm matrices, and we prove Proposition 4.1. From (209) it follows that

[TABLE]

where $||\,.\,||$ denotes the value of the largest element of the matrix in absolute value. From (212) we have

[TABLE]

It follows from Proposition 3.2, (168)–(169), (173) that

[TABLE]

From (255)–(257), it follows that

[TABLE]

We will evaluate ${\rm Re\,}R_{1,11}$ and ${\rm Re\,}R_{2,11}$ to prove Proposition 4.1.

We recall from (173) that $R_{1}$ is a sum of 3 terms. The first term is an integral of $\Delta_{1}^{(1)}$, and the two other terms are integrals of $\Delta_{1}^{(b_{1})}$ and $\Delta_{1}^{(b_{2})}$. We first evaluate the contribution from the terms $\Delta_{1}^{(b_{1})}$ and $\Delta_{1}^{(b_{2})}$.

4.5.1 Contribution to $R_{1}$ from $\Delta_{1}^{(b_{1})}$ and $\Delta_{1}^{(b_{2})}$

It follows from (168) and (169) that

[TABLE]

as $z\to b_{1}$ and $z\to b_{2}$ respectively, and that the matrices $C_{\Delta,1}$ and $C_{\Delta_{2}}$ are given as follows as $s\to\infty$

[TABLE]

We recall that $f$ is real to the main order and from (129), (212) we have

[TABLE]

Since the interior of the bracket $[\,\,]$ in (260) is imaginary to main order, we can calculate the residue in the integral of (259) to find:

[TABLE]

4.5.2 Contribution to $R_{1}$ from $\Delta_{1}^{(1)}$

Denote

[TABLE]

with branch cuts on $J_{2}$ such that the square root is positive as $z\to\infty$. Our goal is to evaluate the terms in (138), and given a matrix $X$ we denote

[TABLE]

Define $D(z)$ and $E^{(\pm)}(z)$ by

[TABLE]

Then from (138) it follows that

[TABLE]

Recalling the definition of $B$ in (127), the definition of $F$ in (128), and the definition of $\widetilde{E}$ in (136), we find that

[TABLE]

We analyze the sign of $\Delta_{1,11}^{(1)}$ in (265). From (129) and (212) we have

[TABLE]

From (263) we see that

[TABLE]

for $e^{i\theta}\in U_{0}\cap C$. Write (265) in the form

[TABLE]

where $\Delta_{1,-j}^{(1)}$ are analytic functions in $z$ in $U_{0}$. Then a calculation of residues gives the following expansion as $z\to 1$:

[TABLE]

We note that $\zeta$ is real on $J_{1}$, and recall the expansion of $\zeta$ in (83)–(84). We also note that ${\rm Im\,}f={\cal O}(s^{2}/n^{2})$, and that $\Phi_{1,11}$ is real but that $\Phi_{1,12}$ and $\Phi_{1,21}$ are imaginary. Combining with (265)–(270), we conclude that

[TABLE]

As a consequence of (262) and (271) we have

[TABLE]

4.5.3 Order of $R_{2}(e^{\pm i\theta_{0}})$

From (266) and (138) we have

[TABLE]

By inspection of the signs of each element, it follows that

[TABLE]

The remaining contributions to $R_{2,11}$, defined in (138), are calculated using rougher estimates from (257). Thus it follows that

[TABLE]

Substituting (272) and (274) into (258) yields Proposition 4.1.

5 Connection to the asymptotics of [7]

Consider the Deift-Its-Zhou asymptotics (6) for 2 fixed gaps $A=(\alpha_{1},\beta_{1})\cup(\alpha_{2},\beta_{2})$. Without loss of generality, we assume that $\nu\equiv\alpha_{2}=-\beta_{1}$. We also denote $\alpha=\alpha_{1}$, $\beta=\beta_{2}$. The following lemma shows that these asymptotics can be extended (with a worse error term) to the region where $\nu$ is decreasing at a sufficiently slow rate as $s\to\infty$. This gives a connection to the asymptotics of Theorem 1.1 (see Remark 1.2 following Theorem 1.1).

Lemma 5.1**.**

Let $\varepsilon>0$. As $s\to\infty$, uniformly for $\nu\in(\nu_{1},\nu_{2})$ where $\nu_{2}>0$ is fixed and $\nu_{1}=\nu_{1}(s)\to 0$ s.t.

[TABLE]

we have

[TABLE]

where $V,\tau,G_{0}$ are defined in equations (12), (11) above with $\nu=\alpha_{2}=-\beta_{1}$, $\alpha=\alpha_{1}$, $\beta=\beta_{2}$; $\gamma=(\beta^{-1}-\alpha^{-1})/8$, and $<x>=x-k$ ($-1/2<<x>\leq 1/2$) with $k$ the closest integer to $x$.

Proof.

Consider the setup of [7] for 2 gaps $(\alpha,-\nu)\cup(\nu,\beta)$. In the notation of [7], $\alpha=a_{0}$, $\nu=-b_{0}=a_{1}$, $\beta=b_{1}$, $s=x$. We now verify that, if $\nu$ tends to zero at a sufficiently slow rate with $s\to\infty$, the jump matrices of the $R$ matrix in the Deift-Its-Zhou RH problem remain uniformly close to the identity, and therefore the analysis of [7] is extendable into that region. We encircle the end-points of the gaps by nonintersecting discs. Note that the discs around $\nu$, $-\nu$ will have to contract as $\nu$ tends to zero, we choose their radia to be $\nu/3$. For the matching of the local parametrices and the global one on the boundaries of the discs, we need, in particular, the parameter (see (4.100), (4.102), etc in [7])

[TABLE]

to be uniformly large in absolute value on the boundary of the disc around $-\nu$. Here $p(z)=(z-\alpha)(z^{2}-\nu^{2})(z-\beta)$, $q(z)=z^{2}+q_{1}z+q_{0}$, where $q_{1}=-(\alpha+\beta)/2$, and the value of the constant $q_{0}$ is determined by the equation

[TABLE]

To analize the integrals in the limit $\nu\to\ 0$, we split the interval $(\nu,\beta)=(\nu,\sqrt{\nu}]\cup[\sqrt{\nu},\beta)$, and change the integration variable $y=t/\sqrt{\nu}$ in the integration over the first one. We then obtain

[TABLE]

where $\gamma=(\beta^{-1}+|\alpha|^{-1})/8$. And therefore, (277) gives

[TABLE]

Substituting this expansion into (276), we obtain that

[TABLE]

on the boundary of the disc around $-\nu$ ($c$ is independent of $z$, $\nu$, $s$). Similarly, we carry out the analysis around the other end-points of the gaps and obtain that the inequality (282) holds for the relevant quantities on the boundaries of all the disc around the end-points of the intervals. In the notation of [7], this means that

[TABLE]

uniformly on the boundaries of the discs.

To prove the lemma, we need to verify that the jump matrices $v_{R,s}(z)$ in (4.123) in [7] have the form $v_{R,s}(z)=I+o(1)$ on the jump contour Figure 4.122 in [7] in the asymptotic regime of the lemma. First, on the boundaries of the discs (see Figure 4.122 in [7]),

[TABLE]

where (see (3.42) in [7])

[TABLE]

Here $\theta(z)=\theta(z;\tau)$ and $V$, $\tau$ are as in (8) and (12),

[TABLE]

The sheet of the Riemann surface $w=p(z)^{1/2}$ is chosen such that $p(z)^{1/2}\to 1$, $z\to\infty$. The constant $u_{\infty}=u(\infty)$, and $d$ is chosen such that the zero of $\theta(u(z)-d)$ coincides with the zero of $\delta(z)-\delta(z)^{-1}$ (which is inside $(-\nu,\nu)$). Note (see [7]) that $\theta(u(z)-d)$ has no other zeros, and $\theta(u(z)+d)$ has no zeros. Thus $f^{\infty}(z)$ is analytic outside $A$ and clearly the limit $f^{\infty}(\infty)=I$. Moreover by standard arguments based on Liouville theorem, $\det f^{\infty}(z)=1$ for all $z$. Furthermore [7], $u_{\infty}+d\equiv 0$ modulo the lattice $m+n\tau$, m,n\in\mbox{\mbox{\msbm Z}}.

In the limit $\nu\to 0$, we have the expansions

[TABLE]

and therefore

[TABLE]

Note that, using the inversion formula ($\tau\to 1/\tau$) for the theta-functions, we can write

[TABLE]

where

[TABLE]

We can now esimate the matrix elements of $f^{\infty}$ on the boundaries of the discs. On the boundary of the disc around $-\nu$, we have $u(z)=-1/2+r(z)$, where $|r(z)|<\varepsilon$ with a suitable $\varepsilon>0$. Recalling periodicity properties of the theta-function, $\theta(z+n\pm\tau)=e^{\mp 2\pi iz-i\pi\tau}\theta(z)$, we write for some $C>0$ uniformly on the boundary

[TABLE]

Similar estimate holds for the other elements of $f^{\infty}$ (we replace $\delta-\delta^{-1}$ and $\theta(u(z)-d)$ in the off-diagonal elements with their derivatives at their zero). In the same vein, using the behaviour of $u(z)$, one obtaines similar estimates on the discs around the other end-points. (Note that, e.g., at $\alpha$, we can assume $|u(z)|<\varepsilon$). Recalling (283) we thus conlude that uniformly on these boundaries

[TABLE]

Adjusting $\varepsilon$, we can write this estimate as $v_{R,s}=I+O((s\nu^{2|<sV>|+\varepsilon})^{-1})$. The error term here is not small at the point $<sV>=1/2$, and we analyse the case of $|<sV>|$ close to $1/2$ separately below. Assume for now that $|<sV>|\leq 1/4$. Then (289) is the estimate we need to prove the lemma in this case. It remains, however, to obtain the same, or better, estimate for $v_{R,s}$ on the intervals outside the discs, where (Figure 4.122 in [7]),

[TABLE]

Since by definition of $q(z)$ ((1.17) in [7] or (10) in the introduction),

[TABLE]

the estimation of $g_{+}(z)$ is similar to that of $\Omega^{(0)}(z)$ above, and we obtain that ${\rm Re\,}(ig_{+}(z))<0$ and

[TABLE]

on the intervals outside the discs, and so

[TABLE]

with some constant $c>0$ independent of $s,\nu,z$. Substituting this into (290), we obtain as above,

[TABLE]

uniformly on the intervals outside the discs. Combining this result with (289), we see that the estimate

[TABLE]

holds uniformly on the whole contour for $R$ in the asymptotic regime of the lemma, and therefore the lemma is proved in the case $|<sV>|\leq 1/4$ by the arguments of [7].

Now consider the remaining case $1/4<|<sV>|\leq 1/2$. Let

[TABLE]

where $k=\pm 1$ is chosen so that $t\in(-1/4,1/4)$. Consider the following function which solves the same jump conditions (given in [7]) as $f^{\infty}$

[TABLE]

Here

[TABLE]

The sheet of the Riemann surface $w=p(z)^{1/2}$ is chosen as before such that $p(z)^{1/2}\to 1$, $z\to\infty$. It is easy to verify that $\gamma(z)-\gamma(z)^{-1}$ has 2 zeros $z_{+}$, $z_{-}$. As $\nu\to 0$, $z_{\pm}=\pm\sqrt{\nu|\alpha\beta|}(1+o(1))$. The constant $d^{\prime}$ is chosen such that the zero of $\theta(u(z)-d^{\prime})$ coincides with the zero $z_{+}$ of $\gamma(z)-\gamma(z)^{-1}$. As in [7], Abel theorem then shows that $u(z_{-})+d^{\prime}\equiv 1/2$ modulo the lattice. Furthermore, $\theta(u(z)-d^{\prime})$ has no other zeros, and $\theta(u(z)+d^{\prime})$ has no zeros. Thus $\widetilde{f}^{\infty}(z)$ is analytic outside $A$ and the limit $\widetilde{f}^{\infty}(z_{-})=I$. It follows by standard arguments that $\det\widetilde{f}^{\infty}(z)=1$ for all $z$. We also note that the limit

[TABLE]

has $\det\Lambda=1$ but is not the identity as before. By standard uniqueness arguments

[TABLE]

The construction of local parametrices $\widetilde{f}_{p}$ around the edge points is similar to that in [7]. The definition of the new $R$-matrix is now as follows: $R(z)=\Lambda f(z)\widetilde{f}_{p}^{-1}(z)$ in the discs around the end-points and $R(z)=\Lambda f(z)(\widetilde{f}^{\infty}(z))^{-1}$ outside. The jump matrices for $R$ at the boundaries of the discs have the same form as before

[TABLE]

and a similar (to the one above) examination of the order of $\widetilde{f}^{\infty}$ on the boundaries shows that uniformly

[TABLE]

As before, a better estimate holds on the rest of the jump contour of $R$. Thus the asymptotics obtained holds in the regime $s\nu^{1/2+\varepsilon}\to\infty$, for $|t|<1/4$. To finish the proof of the lemma it only remains to verify (275) for $|t|<1/4$. By Equation (3.9) in [7],

[TABLE]

where $f_{1}$ is the coefficient in the large $z$ expansion $f(z)=I+f_{1}/z+O(1/z^{2})$. (Below, we also use $f^{\infty}(z)=I+f^{\infty}_{1}/z+O(1/z^{2})$, $\widetilde{f}^{\infty}(z)=\Lambda+\widetilde{f}^{\infty}_{1}/z+O(1/z^{2})$.) By our definition of $R$,

[TABLE]

and therefore, using also (292),

[TABLE]

We have $\Lambda=O(\nu^{-1/4+|t|})$, and since $R_{1}$ has the same order as the error term in (293), $\Lambda^{-1}R_{1}\Lambda=O((s\nu^{1-2|t|+\varepsilon})^{-1})=O((s\nu^{2|<sV>|+\varepsilon})^{-1})$.

Thus

[TABLE]

But it was shown in [7] (Equation (3.48)) that $i(f^{\infty}_{1,22}-f^{\infty}_{1,11})=\frac{\partial}{\partial s}\log\theta(sV;\tau)$, and we again obtain (275) now for $1/4<|<sV>|\leq 1/2$. The lemma is proved.

∎

Remark 5.1**.**

In the overlap region $(0,\nu_{0})\cap(\nu_{1},\nu_{2})$ of the asymptotics of Theorem 1.1 and the lemma, we can explicitly see, as an exercise, the coincidence of the main terms. Indeed, from (11), with $\alpha=\alpha_{1}$, $\beta=\beta_{2}$, $\nu=-\beta_{1}=\alpha_{2}$,

[TABLE]

Substituting here the expansion (281), we obtain

[TABLE]

Since $s\nu\to 0$, we see that this gives exactly the main (order $s^{2}$) term in (24).

Remark 5.2**.**

Integration of the asymptotics of the lemma is related to the determination of the constant $c_{1}$ in (13) which will be addressed in a future publication.

Acknowledgements

We are grateful to Tom Claeys for useful discussions and suggestions. The work of I.K. was partially supported by the Leverhulme Trust research fellowship RF-2015-243.

Appendix

We include a proof of the well-known formula (34), using arguments from [6]. As mentioned in the introduction, the gap probability of $m$ gaps in the bulk scaling limit is given by the sine-kernel Fredholm determinant (2) for a wide class of random matrix ensembles. A particular such ensemble is the Circular Unitary Ensemble (CUE), which is the group of $n\times n$ unitary matrices equiped with the Haar measure. The Haar measure induces a probability measure $p_{n}(\theta)d^{n}\theta$ on the eigenvalues of the matrix given by

[TABLE]

From Heine’s identity and (294), it follows that the probability that there are no eigenvalues on a set $\Sigma\subset C$, where $C$ is the unit circle, is given by the following

[TABLE]

where $D_{n}(J)$ was defined in (14). Denote $J^{(n)}=C\setminus\Sigma^{(n)}$ where

[TABLE]

Using the definition (294) it is easily seen that

[TABLE]

where $\widetilde{K}_{n}(x,y)=\frac{1}{2\pi}\sum_{j=0}^{n-1}e^{ji(x-y)}$. Let

[TABLE]

It follows that

[TABLE]

The kernel $K_{n}$ has the reproducing kernel property, meaning that for $r=1,\dots,n$

[TABLE]

where

[TABLE]

From (295), we see that

[TABLE]

The Fredholm determinant of a trace-class operator $K$ acting on a set $S$ can be represented as

[TABLE]

For bounded $S$ and $K$, one may verify that the sum indeed converges using Hadamard’s inequality. Let $J^{(n)}$ be given by (32), and $\Sigma^{(n)}=C\setminus J^{(n)}$ be the complement. Recall $A$ from (18). Noting (299), we apply (300) to (302) to find that

[TABLE]

where

[TABLE]

For fixed $s$, as $n\to\infty$, we have

[TABLE]

Since the sum (303) converges,

[TABLE]

as $M\to\infty$, for $K=\widehat{K}_{n},K_{s}$, where $s$ remains fixed and uniformly for $n>N$ for some $N$. From (306), it follows that for fixed but arbitrarily large $M$,

[TABLE]

as $n\to\infty$. Thus it follows that

[TABLE]

as $n\to\infty$ and $s$ remains fixed.