The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I

Introduction and statement of results

In this paper we will be concerned with the distribution of the largest eigenvalue $\xi_{\max}{(n)}$ in the following ensembles of matrices $\{M\}$:

The set of $n\times n$ real symmetric matrices, with the probability distribution function (p.d.f.)

The set of $n\times n$ Hermitian matrices, with the p.d.f.

The set of $2n\times 2n$ self-dual Hermitian matrices, with the p.d.f.

Note that in these three external source models, the distributions of eigenvalues of $M$ are unchanged if $\mathbf{A}_{n,\beta}$ (or $\hat{\mathbf{A}}_{n,\beta}$) is changed into $Q\mathbf{A}_{n,\beta}Q^{-1}$ (or $Q\hat{\mathbf{A}}_{n,\beta}Q^{-1}$), where $Q$ is in the orthogonal group $O(n)$, unitary group $U(n)$ and compact symplectic group $Sp(n)$ for $\beta=1,2,4$ respectively. Since we are only concerned with the distribution of eigenvalues of $M$, we assume $\mathbf{A}_{n,\beta}$ (or $\hat{\mathbf{A}}_{n,\beta}$) to be diagonal without loss of generality. To make our presentation uniform for all values of $\beta$, we let $V(x)$ be a fixed function and

be an $n\times n$ diagonal matrix. We assume that $V_{\beta}(x)$ (or $\hat{V}_{\beta}(x)$) and $\mathbf{A}_{n,\beta}$ (or $\hat{\mathbf{A}}_{n,\beta}$) are defined from $V(x)$ and $\mathbf{A}_{n}$ such that

Writing a $2n\times 2n$ self-dual Hermitian matrix into $2\times 2$ blocks $\left(\begin{smallmatrix}a_{st}&b_{st}\\ c_{st}&d_{st}\end{smallmatrix}\right)^{n}_{s,t=1}$, we can express it as a quaternionic Hermitian matrix $(q_{st})^{n}_{s,t=1}$ whose $s,t$-entry comes from the $s,t$-block by

In the quaternion form, the p.d.f. of the Hermitian self-dual external source model is

with $\mathbf{A}_{n}$ defined in (4). The $\mathbf{A}_{n,4}$ defined in (10) corresponds to the $\hat{\mathbf{A}}_{n,4}$ defined in (7). In Appendix A we use the quaternion form p.d.f. (9) of the $4$-external source model to streamline the derivations for all $\beta$.

In this paper we concentrate on the rank $1$ case, i.e.,

Random matrices are powerful tools to simulate Hamiltonians of complex systems. Different types of random matrices, namely the real symmetric (aka orthogonal), the Hermitian (aka unitary) and the Hermitian self-dual (aka symplectic) ensembles are used for physical systems with different properties of time-reversal invariance . For random matrix models without external source in all the three types of ensembles, i.e., rank $\beta$-external source models, the distribution of the largest eigenvalue has been studied extensively for all the three $\beta$. If $\beta=2$, for all real analytic potentials $V_{2}(x)$ under mild regularity conditions, the largest eigenvalue $\xi_{\max}{(n)}$ approaches $\mathbf{e}$, the right-end point of the equilibrium measure of $V_{2}(x)$ (see (16) below), with probability $1$ as $n\to\infty$, and the limiting distribution is the GUE Tracy-Widom distribution. See e.g. and . If $\beta=1$, for real analytic potentials $V_{1}(x)$ satisfying the “one-band” condition (cf. Condition 2 in Subsection 1.2 below) and mild regularity conditions, the largest eigenvalue with probability $1$ approaches $\mathbf{e}$ as $n\to\infty$, and the limiting distribution is the GOE Tracy-Widom distribution. See . If $\beta=4$, similar result can be obtained and the limiting distribution is the GSE Tracy-Widom distribution. See .

The random matrix model with external source was proposed by Brézin and Hikami , to simulate complex systems with both random part and deterministic part. Although in all three types of random matrix ensembles the random matrix model with external source can be defined, due to technical reasons, only the Hermitian ($\beta=2$) type has been studied for general potential functions. See e.g. and references therein.

In , the Hermitian random matrix model with rank $1$ external source was studied for all real analytic potentials $V_{2}(x)$ under mild regularity conditions. For convex potentials, the universality of phase transition was proved. Let $V_{2}(x)$ be defined by $V(x)$ as in (5). In the rank $1$ $2$-external source model, with probability $1$, as $n\to\infty$

where $a$ is the unique nonzero eigenvalue of the external source $\mathbf{A}_{n,2}=\mathbf{A}_{n}$, and $x_{0}(a)$ is a continuous increasing function in $a\in(\frac{1}{2}V^{\prime}(\mathbf{e}),\infty)$ such that $x_{0}(a)\to\mathbf{e}$ as $a\to\frac{1}{2}V^{\prime}(\mathbf{e})$, see (16) and (29). If $a<\frac{1}{2}V^{\prime}(\mathbf{e})$, the limiting distribution of $\xi_{\max}{(n)}$ is the GUE Tracy-Widom distribution, and if $a>\frac{1}{2}V^{\prime}(\mathbf{e})$ the limiting distribution is Gaussian. For the double scaling $a=\frac{1}{2}V^{\prime}(\mathbf{e})+\frac{\alpha}{n^{1/3}}$, the limiting distribution is the generalized Tracy-Widom distribution. If the potential is not convex, then new phenomena may occur. The “critical value” may be less than $\frac{1}{2}V^{\prime}(\mathbf{e})$, and there may be “secondary critical values”. The largest eigenvalue $\xi_{\max}{(n)}$ may converge to two or more points if $a$ takes such values. The results were also obtained by Bertola, Buckingham, Lee and Pierce in and independently.

For real symmetric and Hermitian self-dual matrix models with external source, known results are limited to special potentials. Let $V_{1}(x)$ and $\hat{V}_{4}(x)$ be defined by $V(x)$ as in (5), the rank $1$ $1$-external source model with Gaussian potential ($V(x)=x^{2}$ on the real line) and Laguerre potential ($V(x)=x-c\log(x)$ on half of real line) are studied in e.g. , , and . The limiting location of the largest eigenvalue is given by formula (12), the same as in the corresponding rank $1$ $2$-external source model, where $a$ is the nonzero eigenvalue of $\mathbf{A}_{n}$ and $\mathbf{A}_{n,1}$ is defined by (6). If $a>\frac{1}{2}V^{\prime}(\mathbf{e})$, then the limiting distribution of $\xi_{\max}{(n)}$ is Gaussian, with variance twice of that in the corresponding Hermitian ($\beta=2$) external source model. If $a<\frac{1}{2}V^{\prime}(\mathbf{e})$, the limiting distribution of $\xi_{\max}{(n)}$ is the GOE Tracy-Widom distribution. The rank $1$ $4$-external source model with Laguerre potential is studied in , where the limiting location of the largest eigenvalue is found to be given by formula (12), the same as in the corresponding rank $1$ $2$-external source model, where $a$ is the nonzero eigenvalue of $\mathbf{A}_{n}$ and $\hat{\mathbf{A}}_{n,4}$ is defined by (7). If $a>\frac{1}{2}V^{\prime}(\mathbf{e})$, then the limiting distribution of $\xi_{\max}{(n)}$ is Gaussian, with variance half of that in the corresponding Hermitian ($\beta=2$) external source model. If $a<\frac{1}{2}V^{\prime}(\mathbf{e})$, the limiting distribution of $\xi_{\max}{(n)}$ is the GSE Tracy-Widom distribution. In the limiting distribution of $\xi_{\max}{(n)}$ when $a=\frac{1}{2}V^{\prime}(\mathbf{e})$ is also obtained.

In the recent preprint , Bloemendal and Virág obtained the limiting distribution of the largest eigenvalue $\xi_{\max}{(n)}$ when the potential is Gaussian or Laguerre, for all $\beta$ and for all $a$. When $a$ is at or near $\frac{1}{2}V^{\prime}(\mathbf{e})$, they described the limiting distribution function of $\xi_{\max}{(n)}$ via the unique solution to a PDE. The recent preprint by Mo indicates a new approach to study the limiting distribution of $\xi_{\max}{(n)}$ in the rank $1$ $1$-external source model with Laguerre potential when $a$ is at or near $\frac{1}{2}V^{\prime}(\mathbf{e})$, see also . The contour integral formula in [25, Theorem 1] is equivalent to that of Proposition 2.2 in this paper with $\beta=1$ and Laguerre potential (cf. Remark A.1). In , Mo further simplified the integrand in the contour integral formula, (see [25, Theorem 3],) and he applied it in the asymptotic analysis in to obtain a result similar to that in . In this paper, we take a different approach to apply Proposition 2.2 in asymptotic analysis. The reader may also compare our paper with the paper by Benaych-Georges and Nadakuditi, where they considered a different kind of low rank perturbations of large random matrices.

In this paper, we consider the rank $1$ $\beta$-external source models with general potential $V_{\beta}(x)$ (or $\hat{V}_{\beta}(x)$) which are defined by $V(x)$. The conditions satisfied by $V(x)$ will be given in Subsection 1.2. We find that the “critical value” is independent of $\beta$, and for all $\beta=1,2,4$ find the limiting location of the largest eigenvalue $\xi_{\max}{(n)}$ when $a$, the nonzero eigenvalue of $\mathbf{A}_{n}$, is not equal to the critical value. When $a$ is greater than the critical value, we also find the limiting distribution of $\xi_{\max}{(n)}$.

Besides the asymptotic results summarized above, in Appendix A we also have an algebraic result: the definition of the $\beta$-external source model with general $\beta>0$. Here we note that the analytic method presented in this paper can be used to study the rank $1$ $\beta$-external source model with general $\beta$.

2 Assumptions on V​(x)𝑉𝑥V(x)

Throughout this paper, we assume four conditions on $V(x)$, the function in (5) and (10). The first is

$V(x)$ is a polynomial of degree $2l$ and with positive leading coefficient.

The equilibrium measure $\mu$ associated to $V(x)$ is supported on a single interval $J=[b_{1},b_{2}]$.

For the function $V(x)$ satisfying Conditions 1 and 2, the equilibrium measure $\mu$ has the form

where $\chi_{J}$ is the indicator function and $h(x)$ is a polynomial of degree $2l-2$. The next condition assumed on $V(x)$ is

The function $h(x)$ in the formula (13) of the equilibrium measure $\mu$ of $V(x)$ has only non-real zeros.

The equilibrium measure $d\mu=\Psi(x)dx$ is characterized by the conditions

Conditions 1–3 are assumed to apply Proposition 4.1 in our paper, and they are not used anywhere else in this paper. If Proposition 4.1 can be proved under weaker conditions, e.g. the conditions assumed in [22, Theorem 1] Mariya Shcherbina informed the author that Proposition 4.1 can be proved under the consitions assumed in [22, Theorem 1] through private communication., these conditions can be weakened accordingly.

Functions $V$ satisfying all Conditions 1–4 also satisfy the assumptions of $V$ in [3, Formulas (6)–(8)]. Thus all the results in on $V$ can be applied in this paper.

If $V(x)$ is a convex polynomial with positive leading coefficient, $V(x)$ satisfies Conditions 1–3 by [21, Proposition 3.1], and it is straightforward to verify that $V(x)$ satisfies Condition 4.

3 Preliminary notations

To state the results in this paper, we need a few more notations. We follow the notational convention in to denote the right edge of the support of the equilibrium measure

The so called $\mathbf{g}$-function is defined by

For $a\in(0,\frac{1}{2}V^{\prime}(\mathbf{e}))$, define $c(a)$ as the unique point in $(\mathbf{e},\infty)$ such that

The properties of $\mathbf{g}(x)$ used in this paper is summarized below (see [3, Formula (30)]).

For $a\geq\frac{1}{2}V^{\prime}(\mathbf{e})$, define $c(a):=\mathbf{e}$. We also define two auxiliary functions

The convexity of $\mathbf{g}(x)$ on $(\mathbf{e},\infty)$ yields that for $u>c(a)$,

It is proved in [3, Lemma 1.2] that $\mathcal{A}_{V}$ is an open, semi-infinite interval. From $\mathcal{A}_{V}$ we define

It is also proved in [3, Lemma 1.2] that $\mathbf{a}_{c}>0$.

For $a>\mathbf{a}_{c}$ and $a\not\in\mathcal{J}_{V}$, there is a unique $x_{0}(a)\in(c(a),\infty)$ such that (cf. [3, Lemma 1.3])

For $a>\mathbf{a}_{c}$ and $a\in\mathcal{J}_{V}$, there are $r\geq 2$ and $c(a)<x_{1}(a)<x_{2}(a)<\dots<x_{r}(a)$ such that

We define the set of secondary critical values as $\mathcal{J}_{V}\setminus\{\mathbf{a}_{c}\}$ (cf. [3, Definition 1.3] ).

If the potential $V$ is convex for $x\geq\mathbf{e}$, $\mathcal{J}_{V}=\emptyset$. See [3, Remark 1.2].

4 Statement of main results

Let $V(x)$ be a function that satisfies Conditions 1–4. For any $n$ and $\beta=1,2,4$, let the $n$-dimensional $\beta$-external source models be defined by p.d.f.s (1), (2) and (3) respectively, with potentials $V_{\beta}(x)$ (or $\hat{V}_{\beta}(x)$) given by (5) and external sources $\mathbf{A}_{n,\beta}$ (or $\hat{\mathbf{A}}_{n,\beta}$) given by (4), (6) and (7). We assume that $\mathbf{A}_{n}$ has only one nonzero eigenvalue $a$, as in (11). In each $n$-dimensional $\beta$-external source model, let $\xi_{\max}{(n)}$ be the largest eigenvalue of the random matrix. The theorems below are stated uniformly for all $\beta$-external source models ($\beta=1,2,4$). In the case $\beta=1$, we assume that the dimension $n$ is even. For $\beta=1$ and $n$ is odd, the theorems below also hold, and we discuss it briefly in Appendix A. First we show the limiting location of the largest eigenvalue.

The following hold for each fixed $a$ as $n\to\infty$:

If $0<a<\mathbf{a}_{c}$, or $a=\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and $\mathbf{a}_{c}\notin\mathcal{J}_{V}$, $\xi_{\max}(n)\to\mathbf{e}$ with probability $1$.

If $a>\mathbf{a}_{c}$ and $\mathbf{a}_{c}\not\in\mathcal{J}_{V}$, then $\xi_{\max}(n)\to x_{0}(a)$ with probability $1$, where $x_{0}(a)$ is defined in (29).

If $a>\mathbf{a}_{c}$ and $a\in\mathcal{J}_{V}$, then there exist $r\geq 2$ and $x_{1}(a)$, …, $x_{r}(a)$ defined in (30). Under the assumption that $\mathbf{G}^{\prime\prime}(x_{j}(a))\neq 0$ for all $j=1,\dots,r$, then $\xi_{\max}(n)$ converges to $x_{j}(a)$ with nonzero probability $p_{j,\beta}(0)$ for $j=1,\dots,r$. Here $p_{j,\beta}(0)$ are defined in (168) and $\sum^{r}_{j=1}p_{j,\beta}(0)=1$.

If $a<0$, Theorem 1.1(a) still holds, and the method of proof is similar to that in the $0<a<\mathbf{a}_{c}$ case. Since when $a<0$ there is no interesting phase transition phenomenon for the distribution of the largest eigenvalue (while there is a similar one of the smallest eigenvalue) and the proof is long and parallel to the $a>0$ case, we skip further discussions about the $a<0$ case.

If $a>\mathbf{a}_{c}$, we have the limiting distribution of the largest eigenvalue. If $a$ is not at or near secondary critical values, we have the following result that strengthens Theorem 1.1(b).

The following hold for $a>\mathbf{a}_{c}$ and $a\not\in\mathcal{J}_{V}$ as $n\to\infty$.

where $\Phi(T):=\frac{1}{\sqrt{2\pi}}\int^{T}_{-\infty}e^{-\frac{1}{2}\xi^{2}}d\xi$ denotes the cumulative distribution function of standard normal distribution.

If $a>\mathbf{a}_{c}$ is at or near a secondary critical value, we have the following result that shows the double scaling case and strengthens Theorem 1.1(c).

Suppose that $a_{0}>\mathbf{a}_{c}$ and $a_{0}\in\mathcal{J}_{V}$. Assume that $\mathbf{G}(x;a_{0})$ attains its maximum at $r\geq 2$ points $x_{1}(a_{0})<x_{2}(a_{0})<\dots<x_{r}(a_{0})$ in $(c(a_{0}),\infty)$, and $\mathbf{G}^{\prime\prime}(x_{i}(a_{0});a_{0})\neq 0$ for all $i=1,2,\dots,r$, then for

where $p_{i,\beta}(\alpha)$ ($i=1,\dots,r$) are defined in (168) and $\sum^{r}_{j=1}p_{j,\beta}(\alpha)=1$. Furthermore, $p_{r,\beta}(\alpha)\to 1$ as $\alpha\to\infty$ and $p_{1,\beta}(\alpha)\to 1$ as $\alpha\to-\infty$.

The phenomenon of Theorem 1.3 occurs for some quartic potential $V$ that satisfies Conditions 1–4. For example, $V(x)=0.02093x^{4}-0.16736x^{3}+0.37448x^{2}+0.11418x$.

In the case that $a>\mathbf{a}_{c}$, $a\in\mathcal{J}_{V}$ and $\mathbf{G}^{\prime\prime}(x_{j}(a);a)=0$ at at least one maximizer $x_{j}(a)$ of $\mathbf{G}(x;a)$ in $(c(a),\infty)$, we show hereafter an example when the number of maximizers of $\mathbf{G}(x;a)$ in $(c(a),\infty)$ is $r=2$. The result for general case is similar.

Suppose that $a_{0}>\mathbf{a}_{c}$ and $a_{0}\in\mathcal{J}_{V}$. Assume that $\mathbf{G}(x;a_{0})$ attains its maximum at two points $x_{1}(a_{0})<x_{2}(a_{0})$ in $(c(a_{0}),\infty)$, with $\mathbf{G}^{\prime\prime}(x_{1}(a_{0});a_{0})\neq 0$, $\mathbf{G}^{(2k)}(x_{2}(a_{0});a_{0})\neq 0$ and $\mathbf{G}^{(j)}(x_{2}(a_{0});a_{0})=0$ for $j=1,\dots,2k-1$. Then for

The limiting distribution of the largest eigenvalue when $a\leq\mathbf{a}_{c}$, as well as the limiting location of the largest eigenvalue when $a$ is at or near $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$, will be analyzed in a subsequent paper.

The paper is organized as follows. In Section 2, we calculate the limiting p.d.f. of the largest eigenvalue in the rank $1$ $\beta$-external source model as $n\to\infty$, based on Proposition 2.1. In Section 3, we prove Theorems 1.1, 1.2 1.3 and 1.4. The proof of Proposition 2.1 is in Section 4.

The starting point of the asymptotic analysis in this paper is Proposition 2.2, the contour integral formula of the largest eigenvalue $\xi_{\max}{(n)}$. Since its proof is combinatorial, we postpone it to Appendix A. In this appendix we also propose a definition of the $\beta$-external source model for any $\beta>0$. In Appendix B we show that the results in this paper agree with those in when $\beta=2$.

The p.d.f. of the largest eigenvalue

where $\mathbf{Z}_{m,\beta}$ is the normalization constant. Suppose $F(x_{1},\dots,x_{m})$ is an integrable function with respect to the measure $\mu_{m,\beta}$ defined in (38), define the expectation of $F$ with respect to $\mu_{m,\beta}$ by

Then we can state the technical tool in the asymptotic analysis of this section:

where $R_{\beta}(u)$ is defined in (192).

where $R_{\beta}(u,w_{0})$ is defined in (191).

Let $\epsilon$ be a small positive constant. For all $w$ such that $\operatorname{dist}(w,(-\infty,\mathbf{e}])\geq\epsilon$ and $\lvert w\rvert\leq\epsilon^{-1}$,

where the factor $O(1)$ is bounded uniformly in $w$.

This proposition is a corollary of a theorem of Johansson [21, Theorem 2.4], and we put off its proof to Section 4.

For the asymptotic analysis in this section, we define four types of contours: $\Sigma^{x}_{s_{1},s_{2}}$, $\Pi^{x}_{s}$, $\Gamma^{x}_{s}$ and $I^{x}_{s}$, where $x$ is a real parameter and $s_{1}$, $s_{2}$ and $s$ are positive parameters. We assume $s_{2}>s_{1}/\sqrt{2}$ for $\Sigma^{x}_{s_{1},s_{2}}$ and allow $s=\infty$ in $\Gamma^{x}_{s}$. The contours $\Pi^{x}_{s}$ and $\Sigma^{x}_{s_{1},s_{2}}$ will be used in Subsections 2.1 and 2.2 respectively. The contours $I^{x}_{s}$ and $\Gamma^{x}_{s}$ represent the local parts of $\Pi^{x}_{s}$ and $\Sigma^{x}_{s_{1},s_{2}}$ around the point $x$ respectively, which will turn out to be the saddle point in the asymptotic analysis.

See Figures 4, 4, 4 and 4 for these contours. For any real number $r$, we define

The asymptotic analysis in this section is based on the contour integral representation of the p.d.f. of the largest eigenvalue $\xi_{\max}{(n)}$:

Let $\xi_{\max}{(n)}$ be the largest eigenvalue in the $n$-dimensional rank $1$ $\beta$-external source model for $\beta=1,2,4$, where the potential $V_{\beta}(x)$ (or $\hat{V}_{\beta}(x)$) is defined by (5) from $V(x)$, and the external source matrix $\mathbf{A}_{n,\beta}$ (or $\hat{\mathbf{A}}_{n,\beta}$) is defined by (6) (or (7)) from $\mathbf{A}_{n}$ in (11) with $a>0$. Then for any integer $n$ if $\beta=2,4$ and for even integer $n$ if $\beta=1$,

where $\hat{C}_{n,\beta}$ is a constant, $Z_{n-1,\beta}(u,w)$ is defined in (40), and $\mathcal{C}$ is either $\Sigma^{x}_{s_{1},s_{2}}$ defined in (51) or $\Pi^{x}_{s}$ defined in (52) with $x>u$.

The proof of Proposition 2.2 is in Appendix A.

Let $L$ be a positive number. For sufficiently large $n$, the contour $I^{c(a)}_{L/\sqrt{n}}$ is part of $\Pi^{c(a)}_{\pi_{1}}$. By Proposition 2.1(b), for $w=c(a)+it/\sqrt{n}\in I^{c(a)}_{L/\sqrt{n}}$, the integrand in the contour integral of (56) satisfies

Using the asymptotic formula (57), we have the result

and $\epsilon_{1}(L,n)$ is small if $L$ and $n$ are large. To be precise, for all $\epsilon>0$, there is an $L_{1}>0$ such that for all $L>L_{1}$, $\lvert\epsilon_{1}(L,n)\rvert<\epsilon$ for $n$ large enough.

For $w$ in a bounded subset of $\Pi^{c(a)}_{\pi_{1}}\setminus I^{c(a)}_{L/\sqrt{n}}$, we use Proposition 2.1(c) and estimate the integrand of the contour integral of (56)

If $w(t)\in\Pi^{c(a)}_{\pi_{1}}\setminus I^{c(a)}_{L/\sqrt{n}}$ with $t\geq\pi_{1}$, i.e., $w$ is in the ray from $c(a)+i\pi_{1}$ to $-\infty$, we have

If $\pi_{1}$ is large enough, we have that for all $t\in[\pi_{1},\infty)$ and all $x\in J$, there exists $c_{1}>0$ such that

For $0<t\leq\pi_{1}$, like (62) and (63), we have

where $c_{2}$ is a positive constant depending on $\pi_{1}$.

where the factor $O(1)$ is bounded uniformly in $L$. Substituting the Taylor expansion (210) of $p(x;u,w)$ into (66), we find

By (66) and (67), we find that $\epsilon_{2}(L,n)$ is small if $L$ and $n$ are large. To be precise, for any $\epsilon>0$, there exists $L_{2}>0$ such that for all $L>L_{2}$, $\lvert\epsilon_{2}(L,n)\rvert<\epsilon$ for sufficiently large $n$.

Hence substituting (69) into (40), we have

Similar to (206) and (201), we have (see Remark 4.1)

where $\mathbf{g}(u)$ is defined in (17), $\mathbf{H}(c(a);a)$ is defined in (21), and

Let $L$ be a positive number. For sufficiently large $n$, the contour $\Gamma^{u+n^{-1}}_{L/n}$ is part of $\Sigma^{u+n^{-1}}_{\sigma_{1},\sigma_{2}}$. By Proposition 2.1(a), for $w=u+z/n\in\Gamma^{u+n^{-1}}_{L/n}$ the integrand in the contour integral of (56) satisfies

Using Hankel’s contour integral expression of Gamma function (See [1, 6.1.4]), we find

Comparing the integral on the right-hand side of (80) with the left-hand side of (81), we write analogous to (58) that

and the term $\epsilon_{3}(L,n)$ is small if $L$ and $n$ are large. To be precise, for any $\epsilon>0$, there is an $L_{3}>0$ such that for all $L>L_{3}$, $\lvert\epsilon_{3}(L,n)\rvert<\epsilon$ for sufficiently large $n$.

If $w=w(t)\in\Sigma^{u+n^{-1}}_{\sigma_{1},\sigma_{2}}$ and $0\leq t\leq\sigma_{1}$, i.e., $w$ is in the line segment between $u+n^{-1}$ and $u+n^{-1}+e^{\frac{3\pi i}{4}}\sigma_{1}$, we have

By (24), we know that $\mathbf{H}^{\prime}(u;a)$ is a positive number. If $\sigma_{1}$ is small enough, for all $t\in[0,\sigma_{1}]$ and all $x\in J$

Hence for $0\leq t\leq\sigma_{1}$, substituting (85) in (84), we find with the help of (24)

If $w=w(t)\in\Sigma^{u+n^{-1}}_{\sigma_{1},\sigma_{2}}$ and $t\geq\sigma_{2}+(1-\sqrt{2}/2)\sigma_{1}$, i.e., $w$ is in the ray from $u+n^{-1}-\sigma_{1}/\sqrt{2}+i\sigma_{2}$ to $-\infty$, like (62) we have

If $\sigma_{2}$ is large enough, like (63) we have that for all $t\in[\sigma_{2}+(1-\sqrt{2}/2)\sigma_{1},\infty)$ and all $x\in J$, if $n$ is large enough

Substituting (88) into (87), we find that like (88), for $t\geq\sigma_{2}+(1-\sqrt{2}/2)\sigma_{1}$

If $w=w(t)\in\Sigma^{u+n^{-1}}_{\sigma_{1},\sigma_{2}}$ and $\sigma_{1}\leq t\leq\sigma_{2}+(1-\sqrt{2}/2)\sigma_{1}$, i.e., $w$ is in the line segment between $u+n^{-1}+e^{\frac{3\pi i}{4}}\sigma_{1}$ and $u+n^{-1}-\sigma_{1}/\sqrt{2}+i\sigma_{2}$ we have

where the last factor $O(1)$ is bounded uniformly in $L$. Substituting the Taylor expansion (207) of $p(x;u,w)$ into (61), we find that like (67)

By (91) and (92), we find that $\epsilon_{4}(L,n)$ is small if $L$ and $n$ are large. To be precise, for any $\epsilon>0$, there is an $L_{4}>0$ such that for all $L>L_{4}$, $\lvert\epsilon_{4}(L,n)\rvert<\epsilon$ for sufficiently large $n$.

where $\mathbf{G}(u;a)$ is defined in (20), $\mathcal{M}_{\beta}(u)$ is defined in (83) and $\mathbf{C}_{n,\beta}$ is defined in (78).

The inequality (98) is a straightforward consequence of the definitions (40) and (41) of $Z_{n-1,\beta}(u,w)$ and $\hat{Z}_{n-1,\beta}(u,w)$.

Like (64), we have that for $\pi_{2}$ large enough, for all $t>\pi_{2}$ and $x\in J$, there exists $c^{\prime}_{1}>0$ such that (cf. (63) and (64))

For all $0<t\leq\pi_{1}$ and $x\in J$, we have like (65) and (90) that

On the other hand, we assume that $\pi_{4}$ is large enough such that for all $x<\mathbf{e}$,

Using the estimate (116) and (117) of $Z_{n-1,\beta}(u,w)$, we find by direct calculation

Thus similar to (116), we have by (122) that for $w\in\Pi^{u+1}_{1}$

Using the estimates (123) and (124) of $Z_{n-1,\beta}(u,w)$, we have like (118)

Proofs of Theorems 1.1, 1.2, 1.3 and 1.4

In this section we prove the main theorems in this paper. We divide the proofs into three subsections. In Subsection 3.1, we consider the case that $0<a<\mathbf{a}_{c}$ and the case that $a=\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and $\mathbf{a}_{c}\notin\mathcal{J}_{V}$, and prove Theorem 1.1(a). In Subsection 3.2, we consider the case that $a>\mathbf{a}_{c}$ and $a\not\in\mathcal{J}_{V}$, and prove Theorems 1.2 and 1.1(b). In Subsection 3.3, we consider the case that $a>\mathbf{a}_{c}$ and $a\in\mathcal{J}_{V}$ and prove Theorems 1.3, 1.1(c) and 1.4.

First we consider the case that $0<a<\mathbf{a}_{c}$. Let $\epsilon$ be a small positive number, such that $\mathbf{e}+\epsilon<c(a)$ and $\mathbf{e}+\epsilon^{-1}>c(a)$. Furthermore we assume that $\epsilon$ is small enough such that the inequalities (127), (129) and (131) hold.

The condition $0<a<\mathbf{a}_{c}$ implies the inequality $\mathbf{H}(c(a);a)>\mathbf{G}_{\max}(a)$, see (25)–(30) and [3, Lemma 1.2(d)]. We assume that

We assume that for $u,v\in[c(a)-\epsilon,c(a)+\epsilon]$

We assume that for $u\geq\mathbf{e}+\epsilon^{-1}$,

By (128), (130), (132), (134), we find that

Let $\epsilon^{\prime}<\epsilon/2$ be a small positive number such that

The probabilities (135), (136) and (137) imply that the conditional probability

Since $[\mathbf{e}+\epsilon^{\prime},\mathbf{e}+\epsilon^{\prime}]\cup(\mathbf{e}-\epsilon,\mathbf{e}+2\epsilon^{\prime})\in[\mathbf{e}-\epsilon,\mathbf{e}+\epsilon]$, (138) implies that

Taking $\epsilon$ arbitrarily small, we prove Theorem 1.1(a) when $0<a<\mathbf{a}_{c}$.

The case when $a=\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and $\mathbf{a}_{c}\notin\mathcal{J}_{V}$ is similar. Let $\epsilon$ be a small enough positive number. Since $a=\frac{1}{2}V^{\prime}(\mathbf{e})$, we have $c(a)=\mathbf{e}$. Since $a=\mathbf{a}_{c}\notin\mathcal{J}_{V}$, there exists $\epsilon^{\prime}>0$ depending on $\epsilon$ such that for all $u>\mathbf{e}+\epsilon$, $\mathbf{G}(u;a)<\mathbf{H}(\mathbf{e};a)-\epsilon^{\prime}$. Thus like (128), for $u\in[c(a)+\epsilon,\mathbf{e}+\epsilon^{-1}]$ we have

When $\epsilon$ is small enough, (132) and (134) also hold. Then by arguments similar to (135)–(139), we prove Theorem 1.1(a) when $a=\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and $\mathbf{a}_{c}\notin\mathcal{J}_{V}$.

Let $\epsilon$ be a small enough positive constant such that the maximizer $x_{0}(a)$ of $\mathbf{G}(x;a)$ in $[c(a),\infty)$ is less than $\mathbf{e}+\epsilon^{-1}$, and the inequalities (141), (145) and (147) are satisfied.

First we consider the case that $\mathbf{a}_{c}<a<\frac{1}{2}V^{\prime}(\mathbf{e})$, i.e., $c(a)>\mathbf{e}$. We assume that

We assume that for all $u\in[\mathbf{e}-\epsilon,\mathbf{e}+\epsilon]$

We assume that for all $u\in[c(a)-\epsilon,c(a)+\epsilon]$

We assume that for $u\geq\mathbf{e}+\epsilon^{-1}$,

By (142), (144), (146), (148) and (150), we find that

In the case that $a\geq\frac{1}{2}V^{\prime}(\mathbf{e})$, i.e., $c(a)=\mathbf{e}$, we find that inequalities (144), (148) and (150) still hold, and the estimate (151) holds with $c(a)=\mathbf{e}$.

For $u\in[c(a)+\epsilon,\mathbf{e}+\epsilon^{-1}]$, we have the asymptotic formula (97) and $x_{0}(a)$ is the unique maximum of $\mathbf{G}(x;a)$ in $[c(a)+\epsilon,\mathbf{e}+\epsilon^{-1}]$. If we further assume that $\mathbf{G}^{\prime\prime}(x_{0}(a);a)\neq 0$, by the standard Laplace’s method we have that

The probabilities (151), (152) and (153) imply Theorem 1.2(a).

If the second derivative of $\mathbf{G}(x;a)$ vanishes at $x_{0}(a)$, due to the analyticity of $\mathbf{G}(x;a)$, there exists $k>1$ such that $\mathbf{G}^{(j)}(x_{0}(a);a)=0$ for $j=1,\dots,2k-1$ and $\mathbf{G}^{(2k)}(x_{0}(a);a)\neq 0$. By the Laplace’s method we have

The probabilities (151), (154) and (155) imply Theorem 1.2(b). Finally, Theorem 1.2 implies Theorem 1.1(b).

Let $a_{0}>\mathbf{a}_{c}$, and $\epsilon$ be a small positive constant such that the inequalities (142), (144), (146), (148) and (150) hold with $a=a_{0}$. It is easy to verify that there exists a positive number $\bar{\epsilon}$ depending on $\epsilon$ such that if we take $a=a_{0}+\epsilon^{\prime}$ with $\epsilon^{\prime}\in[-\bar{\epsilon},\bar{\epsilon}]$, the inequalities (142), (144), (146), (148) and (150) still hold with the same $\epsilon$. Thus the estimate of probability (151) still holds with $a=a_{0}+\epsilon^{\prime}$. If we further assume that

First we assume that $a_{0}\in\mathcal{J}_{V}$ and $\mathbf{G}(x;a_{0})$ has $r\geq 2$ maximizers $x_{1}(a_{0})<x_{2}(a_{0})<\dots<x_{r}(a_{0})$ in $(c(a_{0}),\infty)$, and all of them are less than $\mathbf{e}+\epsilon^{-1}$. Further we assume that for all $i=1,\dots,r$

For $x$ around $x_{i}(a_{0})$ ($i=1,\dots,r$), we denote

Let $\epsilon_{i}$ ($i=1,\dots,r$) be small enough constant numbers such that $x_{i}(a_{0})$ is the unique maximum of $\mathbf{G}(x;a_{0})$ in $[x_{i}(a_{0})-\epsilon_{i},x_{i}(a_{0})+\epsilon_{i}]$. Applying the standard Laplace’s method to (97), near $x_{i}(a_{0})$ ($i=1,2$), we obtain that

There exists $\epsilon^{\prime\prime}>0$ depending on $\epsilon_{1},\dots,\epsilon_{r}$ such that for sufficiently large $n$

From (168) we immediately find $\sum^{r}_{j=1}p_{j,\beta}(\alpha)=1$. By (167) and (168) we find that $\lim_{\alpha\to\infty}p_{r,\beta}(\alpha)=1$ and $\lim_{\alpha\to-\infty}p_{1,\beta}(\alpha)=1$.

Therefore Theorem 1.3 is proved. Theorem 1.1(c) is a consequence of Theorem 1.3 with $\alpha=0$.

Next we consider the case that $r=2$ and for $k>1$

Let $\epsilon_{i}$ ($i=1,2$) be small enough constant numbers such that $x_{i}(a_{0})$ is the unique maximum of $\mathbf{G}(x;a_{0})$ in $[x_{i}(a_{0})-\epsilon_{i},x_{i}(a_{0})+\epsilon_{i}]$. Applying the standard Laplace’s method to (97), near $x_{1}(a_{0})$, we obtain similar to (163) and (164) that

Applying the Laplace’s method to (97), near $x_{2}(a_{0})$, we obtain

Also there exists $\epsilon^{\prime\prime}>0$ depending on $\epsilon_{1}$ and $\epsilon_{2}$ such that the estimate (166) holds. The probabilities (158), (166), (177), (178), (179), (180) show that the probability that $\xi_{\max}{(n)}$ is in $[x_{1}-\epsilon_{1},x_{1}+\epsilon_{1}]$ or $[x_{2}-\epsilon_{2},x_{2}+\epsilon_{2}]$ approaches $1$ as $n\to\infty$.

Proof of Proposition 2.1

The proof of Proposition 2.1 is based on a theorem of Johansson . For the convenience of readers we state it bellow.

[21, Theorem 2.4] Suppose the function $V$ satisfies Conditions 1–3, and $\mu$ is the equilibrium measure associated to $V$. Let $f$ be a real function that satisfies conditions (i)–(iii) below, with $s=2$ if $\beta=2$ and $s=17/2$ if $\beta\neq 2$. Then there are a quadratic functional $A$ on $f$ and a signed measure $\nu$ on $\operatorname{supp}(\mu)=J=[b_{1},b_{2}]$ which do not depend on $n$, such that as $n\to\infty$

The quadratic functional $A$ is defined by

From the quadratic functional $A$, we define the inner product $\langle\cdot,\cdot\rangle_{A}$ by

The explicit formula of $\nu(x)$ is more complicated and is given in [21, Formula (3.54)]. The conditions mentioned in Proposition 4.1 are (see [21, Page 157])

For any $x_{0}>0$, there is an $\alpha>0$ such that $h\psi_{x_{0}}\in H^{s+\alpha}$, where $H^{s}$, $s>0$, is the standard $L^{2}$ Sobolev space, and $\psi_{x_{0}}\in C^{\infty}$ is the function such that $\psi_{x_{0}}(x)=1$ if $\lvert x\rvert\leq x_{0}$, $\psi_{x_{0}}(x)=0$ if $\lvert x\rvert\geq x_{0}+1$ and $0\leq\psi_{x_{0}}(x)\leq 1$.

The function $R_{\beta}(u,w)$ appearing in Proposition 2.1 is defined by

To facilitate the proof of Proposition 2.1, we define

where $c$ is a parameter no greater than $u$. When $c=u$, $Z_{m,\beta}(u,w;c)$ and $\hat{Z}_{m,\beta}(u,w;c)$ become $Z_{m,\beta}(u,w)$ and $\hat{Z}_{m,\beta}(u,w)$.

The proof of Proposition 2.1 is as follows. Recall that $\mathbf{e}$ is the right edge of $J$, the support of the equilibrium measure. Let $\bar{c}=(\mathbf{e}+u)/2$. We write

In case (a) where $w$ is given by (46), we assume the results

and find that (47) is the consequence of (196), (197), (198) and (199).

In case (b) where $w$ is given by (48), we assume the results

We still have (199), and (49) is the consequence of (200), (201), (202) and (199).

Below we prove the asymptotic formulas (196), (197), (198), (200), (201) and (203).

In the proof, $c$ stands for $\bar{c}=(\mathbf{e}+u)/2$ or $u$.

$f_{1}(x)$ and $f_{2}(x)$ satisfy the conditions (i)–(iii) mentioned in Proposition 4.1.

$f_{1}(x)=f_{2}(x)=p(x;u,w)$ for $x\leq\mathbf{e}$.

$f_{1}(x)\geq p(x;u,w)$ for $x\in(\mathbf{e},c)$.

There exists $x_{0}\in(\mathbf{e},c)$ such that $f_{2}(x)\leq p(x;u,w)$ for $x\in(\mathbf{e},x_{0})$ and $f_{1}(x)-f_{2}(x)\geq\log 2$ for $x\geq x_{0}$.

As a consequence of the properties of $f_{1}(x)$ and $f_{2}(x)$, we have

By Proposition 4.1, we have for both $i=1,2$ that

Thus by the sandwich inequality (204) and (194) we obtain

By (206), we complete the proof of (203) with $c=u$. Let $w$ be given in (46), we have uniformly for all $x\leq\mathbf{e}$ that

and we obtain the proof of (197) with $c=\bar{c}$ and the proof of (196) with $c=u$. Let $w$ be given in (48), we have uniformly for all $x\leq\mathbf{e}$ that

and we obtain the proof of (201) with $c=\bar{c}$ and the proof of (200) with $c=u$. ∎