On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank one case

Introduction and results

Fix an $n\times n$ Hermitian matrix $\mathbf{A}_{n}$ and consider the following density function on the set $\mathcal{H}_{n}$ of $n\times n$ Hermitian matrices:

where $Z_{n}$ is the normalization constant. Here the ‘external potential’ $V(x)$ is a real-valued function which decays fast enough as $|x|\to\infty$ so that $Z_{n}$ is convergent. The matrix $\mathbf{A}_{n}$ is called the external source: see e.g. , , , , , , . Note that the distribution of eigenvalues of $M$ is unchanged if $\mathbf{A}_{n}$ is replaced by $U\mathbf{A}_{n}U^{-1}$ for any unitary matrix $U$. Since we are only concerned on eigenvalues of $M$, we assume without loss of generality that $\mathbf{A}_{n}$ is a diagonal matrix.

A special case is when for all $n$, the external source has a fixed number $\mathbf{m}$, called the rank of $\mathbf{A}_{n}$, of fixed non-zero eigenvalues. In this case, the sequence of probability spaces $(\mathcal{H}_{n},p_{n})$ is called a Hermitian matrix model with spiked external source, spiked source model for short. In this paper we only consider the case when $\mathbf{m}=1$. The higher rank case when $\mathbf{m}>1$ will be analyzed in the upcoming companion paper. Throughout this paper, we assume that $n\geq 1$ and

where $a$ is a real number, independent of $n$.

There are two important special cases. When $V(x)=x^{2}/2$, the spiked source model is called the GUE spiked model. The density $p_{n}(M)$ is that of $M=H+\mathbf{A}_{n}$ where $H$ is an $n\times n$ GUE (Gaussian unitary ensemble) matrix. When $V(x)=((1+c)x-c\log x)\chi_{(0,\infty)}(x)$, $c=(m-n)/n\geq 0$, the spiked source model is the complex Wishart spiked model. In this case, setting $\Sigma:=(1-(1+c)^{-1}\mathbf{A}_{n})^{-1}$, the density $p_{n}(M)$ is that of $M=\Sigma^{1/2}XX^{\dagger}\Sigma^{1/2}$ where $X$ is an $n\times m$ complex rectangular matrix with i.i.d. standard complex Gaussian entries.For the complex Wishart spiked model, $V$ is not real analytic at $x=0$. Throughout this paper, we only consider $V$ which is real analytic in the whole line. However, the method can be generalized to the Wishart-type potentials in a straightforward way. For these two cases, the limit of the largest eigenvalue $\xi_{\max}(n)$ of $M$ was studied in great detail in and . An important feature is the following phase transition phenomenon. Let $\mathbf{e}$ denote the right-end point of the limiting empirical distribution of the eigenvalues of the Hermitian matrix model with no external source (see (10) below).The limiting empirical distribution in the spiked source model is the same as the Hermitian model with no external source. It was shown in both the GUE and the complex Wishart spiked models that as $n\to\infty$, with probability $1$,

Here the function $G(T)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{T}e^{-\frac{1}{2}\xi^{2}}d\xi$ is the cumulative distribution function of the standard normal distribution, and $F_{0}$ and $\sqrt{F_{1}}$ are the GUE and GOE Tracy-Widom distribution functions, respectively. They are defined in (16) and (20) below, respectively. A limit theorem was also proven for the double scaling case when $a=\frac{1}{2}V^{\prime}(\mathbf{e})+\frac{\alpha}{n^{1/3}}$.

The purpose of this paper is to extend the results (3)–(5) to the spiked source model with general potential $V$. It turns out that if $V(x)$ is convex in the interval $x\in(\mathbf{e},\infty)$, then all of (3)–(5) still hold. Especially, the ‘critical value’ of $a$ is again given by $\frac{1}{2}V^{\prime}(\mathbf{e})$. However, if $V$ is not convex in $(\mathbf{e},\infty)$, new features may occur. Two key new features are the followings.

The critical value of $a$ may be smaller than $\frac{1}{2}V^{\prime}(\mathbf{e})$. See Lemma 1.2 and Theorem 1.2. For such a case, when $a$ equals this critical value, $\xi_{\max}(n)$ does not converge with probability 1. Instead it converges to two or more values, each with non-zero probability. In this case, the fluctuation of $\xi_{\max}(n)$ is generically $F_{0}$ at the smallest limiting value and $G$ at the larger limiting values. See Theorem 1.3.

There may be a discrete set of ‘secondary critical values’ of $a$, which are greater than the critical value. If $a$ is at a secondary critical value, then $\xi_{\max}(n)$ converge to two or more values, each with non-zero probability. In this case, the fluctuation of $\xi_{\max}(n)$ is generically $G$ at each of the limiting values. See Theorem 1.4.

The exact assumptions on the potential $V$ is given in Subsection 1.2. The universality result for convex potentials is in Subsection 1.3. In Subsection 1.4 we define the critical and the secondary critical values for non-convex potentials. The limit laws for the non-convex potentials are given in Subsection 1.5.

While we were preparing for this paper and the companion paper for the higher rank case, we learned that M. Bertola, R. Buckingham, S. Y. Lee and V. Pierce were also working on the spiked source models (see for the first part of their work). While we focus, especially in the second paper, on the limit laws when $a_{1},\cdots,a_{\mathbf{m}}$ are distinct, Bertola, Buckingham, Lee and Pierce focus on the case when $a_{1}=\cdots=a_{\mathbf{m}}$ and $\mathbf{m}\to\infty$ slower than $n$. Also we use the asymptotics of usual orthogonal polynomials but Bertola, Buckingham, Lee and Pierce use asymptotics of multiple orthogonal polynomials via Riemann-Hilbert problem of size larger than $2$.

Before closing this subsection, we mention that the spiked real symmetric matrix model is much more difficult. Even for the GOE and the real Wishart case, the limiting distribution at the critical value is not yet known. For the quaternionic case, the limiting distribution is obtained when the rank $\mathbf{m}=1$ (see for the Wishart model; Gaussian model is also similar).

We also mention that there are several results for the spiked Wigner ensembles and spiked sample covariance matrices. See, for example, , , , , , and .

2 Assumptions on external potential V𝑉V.

Throughout this paper, we assume the following three conditions on $V$:

The second condition is to ensure the convergence of the density function: compare this with the condition on $V$ in . The third condition on being ‘regular’ is a technical condition as defined in . We need a few definitions to state it.

First, recall the equilibrium measure and the so-called $\mathbf{g}$-function. General references are and . For a given potential $V$, the empirical distribution of the eigenvalues of the matrix model with no external source converges to the associated equilibrium measure $\mu$. The equilibrium measure is characterized by a certain variational problem. If $V$ is real analytic, $\mu$ is supported on a finite union of intervals,

for some $N\geq 0$. We denote the right-most edge of the support by

On $J$, $d\mu$ has the form $d\mu=\Psi(x)dx$,

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

The potential $V$ is said to be regular (see ) if

The first condition implies that the function $\Psi(x)>0$ for all $x\in J$, and also that $\Psi(x)$ vanishes like a square-root at each end of the interval of the support. This in turn implies, in particular, that for the model with $\mathbf{A}_{n}=0$, the largest eigenvalue has the limiting distribution given by $F_{0}$ (see e.g. for the non-varying weight; varying weight case is similar using the analysis of .) Note that the second condition restricted to the domain $x>\mathbf{e}$ implies that

3 Statement of results: convex potentials

Let $F_{0}(T)$ be the GUE Tracy-Widom distribution defined by

where $\chi_{[T,\infty)}$ denotes the projection operator on $[T,\infty)$, and $K_{\operatorname{Airy}}$ is the Airy operator defined by the kernel

Here $\operatorname{Ai}$ is the Airy function.

where the contour is from $\infty e^{5\pi i/6}$ to $\infty e^{\pi i/6}$ and the pole $z=-i\alpha$ lies above the contour in the complex plane: see Figure 1.

where $\langle f,g\rangle_{E}$ denotes the real inner product over $E$, $\int_{E}f(x)g(x)dx$. (See [3, Definition 1.3].) When $\alpha=0$,

equals the square of the GOE Tracy-Widom distribution (see [3, Formula (24)]).

Fix a potential $V$ satisfying the assumptions (6)–(8). In the companion paper on the higher rank case, we need to consider the spiked source model of rank $1$ whose density function is same as in (1) but with the change that the matrix $M$ is now of size $n-j+1$ and $\mathbf{A}_{n}$ is replaced by $\mathbf{A}_{n-j+1}$ for fixed $j$:

Let $\mathbf{e}$ be as in (10). Set (recall (11))

for $x_{*}>\mathbf{e}$, if $V^{\prime\prime}(x_{*})-\mathbf{g}^{\prime\prime}(x_{*})>0$. Note that if $V(x)$ is convex in $x\geq\mathbf{e}$, then $V^{\prime\prime}(x)-\mathbf{g}^{\prime\prime}(x)>0$ for all $x>\mathbf{e}$. For later reference we note that $V^{\prime\prime}(x_{*})-\mathbf{g}^{\prime\prime}(x_{*})=-\mathbf{G}^{\prime\prime}(x_{*})$ in terms of the notation (32) that is defined below.

The following is the first main result of this paper. Let $V(x)$ be a potential that is convex in $x\in(\mathbf{e},\infty)$. For $a>\frac{1}{2}V^{\prime}(\mathbf{e})$, let $x_{0}(a)$ be the unique maximizer of the function $\mathbf{g}(x)-V(x)+ax$ in $x\in(\mathbf{e},\infty)$. Such a maximizer exists since $\mathbf{g}(x)-V(x)$ in $x\in(\mathbf{e},\infty)$ is strictly concave and $\mathbf{g}^{\prime}(\mathbf{e})-V^{\prime}(\mathbf{e})+a=-\frac{1}{2}V^{\prime}(\mathbf{e})+a>0$ (see (30)) and $\mathbf{g}^{\prime}(x)-V^{\prime}(x)+a<0$ for all large enough $x$. This $x_{0}(a)$ is same as in Lemma 1.3.

Let $V(x)$ be a potential that is convex in $x\in(\mathbf{e},\infty)$. Set

Hence the transition phenomenon is universal for convex potentials. The next two subsections are about non-convex potentials

4 Critical value and secondary critical values

In this Subsection, we define critical values and the secondary critical values of $a$.

By definition (14), $\mathbf{g}(x)$ is real analytic in $(\mathbf{e},\infty)$, is continuously differentiable in $[\mathbf{e},\infty)$ and satisfies

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

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

Note that $c(a)$ decreases strictly in $a\in(0,V^{\prime}(\mathbf{e})/2)$ and continuous in $a\in(0,\infty)$.

Let $a>0$. We have the following properties.

$\mathbf{H}(x)$ is a convex function in $x\in[\mathbf{e},\infty)$ with the unique minimum attained at $x=c(a)$.

$\mathbf{H}(x)>\mathbf{G}(x)$ for all $x\in(\mathbf{e},\infty)$.

$\displaystyle\lim_{x\downarrow\mathbf{e}}\mathbf{H}^{\prime}(x)=\lim_{x\downarrow\mathbf{e}}\mathbf{G}^{\prime}(x)=a-\frac{1}{2}V^{\prime}(\mathbf{e})$.

As $x\to+\infty$, $\mathbf{H}(x)\to+\infty$, $\mathbf{H}(x)/x\to a$, $\mathbf{G}(x)\to-\infty$ and $\mathbf{G}(x)/x\to-\infty$.

See Figures 2, 3 and 4 for a few examples of the graphs of $\mathbf{G}$ and $\mathbf{H}$.

The critical value for the spiked source model with potential $V$ is defined as

$(\frac{1}{2}V^{\prime}(\mathbf{e}),\infty)\subset\mathcal{A}_{V}$. Hence $\mathbf{a}_{c}\leq\frac{1}{2}V^{\prime}(\mathbf{e})$.

The set $\mathcal{A}_{V}$ is an open, semi-infinite interval. Hence $\mathcal{A}_{V}=(\mathbf{a}_{c},\infty)$.

For $0<a<\mathbf{a}_{c}$, we have $\mathbf{G}(x;a)<\mathbf{H}(c(a);a)$ for all $x\in(c(a),\infty)$.

If the potential $V(x)$ is convex for $x\geq\mathbf{e}$, then $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$, and $\mathbf{G}(x;\mathbf{a}_{c})<\mathbf{H}(\mathbf{e};\mathbf{a}_{c})$ for all $x>\mathbf{e}$. (Note that $c(\mathbf{a}_{c})=\mathbf{e}$ and $\mathbf{G}(\mathbf{e};\mathbf{a}_{c})=\mathbf{H}(\mathbf{e},\mathbf{a}_{c})$.)

If the potential $V$ is such that $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$, then $\mathbf{G}(x;\mathbf{a}_{c})\leq\mathbf{H}(c(\mathbf{a}_{c});\mathbf{a}_{c})$ for all $x\in(c(a),\infty)$, and the equality is attained at least at one point.

Let $a\in(\frac{1}{2}V^{\prime}(\mathbf{e}),\infty)$. Since $\lim_{x\downarrow\mathbf{e}}\mathbf{G}^{\prime}(x)=a-\frac{1}{2}V^{\prime}(\mathbf{e})>0$, there is $\bar{x}>\mathbf{e}$ such that $\mathbf{G}(\bar{x})>\mathbf{G}(\mathbf{e})=\mathbf{H}(\mathbf{e})$. Thus $a\in\mathcal{A}_{V}$.

The continuity of $\mathbf{G}$ and $\mathbf{H}$ in $a$ implies that $\mathcal{A}_{V}$ is an open set. Now we show that $\mathcal{A}$ is a semi-infinite interval. Suppose that $a\in\mathcal{A}_{V}$ and $a<\frac{1}{2}V^{\prime}(\mathbf{e})$. Let $\bar{x}\in(c(a),\infty)$ be the point such that $\mathbf{G}(\bar{x};a)>\mathbf{H}(c(a);a)$. Let $a^{\prime}\in(a,\frac{1}{2}V^{\prime}(\mathbf{e})]$. From Definition 1.1 of $c(a)$, we see that $c(a^{\prime})<c(a)$, and hence $\bar{x}\in(c(a^{\prime}),\infty)$. Moreover,

is strictly positive since each term in bracket is strictly positive. Thus $a^{\prime}\in\mathcal{A}_{V}$, and this, together with (a), implies that $\mathcal{A}_{V}$ is a semi-infinite interval.

Let $0<a<\mathbf{a}_{c}$. Suppose that there is $\bar{x}\in(c(a),\infty)$ such that $\mathbf{G}(\bar{x};a)=\mathbf{H}(c(a);a)$. For any $a^{\prime}\in(a,\mathbf{a}_{c})$, we have $c(a)>c(a^{\prime})$ since $a<a^{\prime}<\frac{1}{2}V^{\prime}(\mathbf{e})$. Thus we find from (35) that $\mathbf{G}(\bar{x};a^{\prime})-\mathbf{H}(c(a^{\prime});a^{\prime})>0$. This implies that $a^{\prime}\in\mathcal{A}_{V}$ which is a contradiction.

Let $0<a<\frac{1}{2}V^{\prime}(\mathbf{e})$. We will show that $a\notin\mathcal{A}$. Since $V$ is convex, $\mathbf{G}(x)$ is concave in $x\in(\mathbf{e},\infty)$. As $\mathbf{G}^{\prime}(\mathbf{e};a)<0$, this implies that $\mathbf{G}(x)$ is decreasing in $x\in(\mathbf{e},\infty)$. Thus for $x\in(c(a),\infty)$, $\mathbf{G}(x)<\mathbf{G}(c(a))<\mathbf{H}(c(a))$. Hence $a\notin\mathcal{A}_{V}$. When $a=\frac{1}{2}V^{\prime}(\mathbf{e})$, a similar argument implies that $\mathbf{G}(x)<\mathbf{H}(x)$ for all $x>\mathbf{e}$.

This follows from the continuity of $\mathbf{G}$ and $\mathbf{H}$ in $a$ and the fact that $\mathbf{a}_{c}=\inf\mathcal{A}_{V}$.

See typical graphs of $\mathbf{G}$ and $\mathbf{H}$ for $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ in Figure 2, and typical graphs of $\mathbf{G}$ and $\mathbf{H}$ for $0<\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$ in Figure 3.

When $V$ is non-convex, there may exist $\bar{x}>\mathbf{e}$ such that $\mathbf{G}(\bar{x};\mathbf{a}_{c})=\mathbf{H}(\mathbf{e},\mathbf{a}_{c})$ even if $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$.

By Definition 1.2 of $\mathbf{a}_{c}$, when $a>\mathbf{a}_{c}$, $\mathbf{G}(x;a)>\mathbf{H}(c(a);a)$ for some $x>c(a)$. The point $x$ at which $\mathbf{G}(x;a)$ attains its maximum plays an important role. Indeed, we will show in the below that if the maximum is attained at a unique point, then $\xi_{\max}(n)$ converges to this point (see Theorem 1.2). However, it may happen that for some $a$’s, the function $\mathbf{G}(x;a)$ attain its maximum at more than one point. Let

This set is discrete since $\mathbf{G}(x;a)$ is analytic in both $x$ and $a$. Note that when $V$ is convex, $\mathbf{a}_{c}\notin\mathcal{J}_{V}$ from Lemma 1.2(e). For a non-convex $V$, as indicated in Remark 1.1, $\mathbf{a}_{c}$ may or may not be in $\mathcal{J}_{V}$ no matter if $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$. See typical graphs of $\mathbf{G}$ and $\mathbf{H}$ for $a\in\mathcal{J}_{V}$ in Figure 4.

For $a\in[\mathbf{a}_{c},\infty)$ such that $a\not\in\mathcal{J}_{V}$, let $x_{0}(a)$ be the unique point in $[c(a),\infty)$ at which $\mathbf{G}(x;a)$ attains its maximum. Then $x_{0}(a)$ is a continuous, strictly increasing function in $a\in[\mathbf{a}_{c},\infty)\setminus\mathcal{J}_{V}$.

If $a_{0}\in\mathcal{J}_{V}$ and $a_{0}>\mathbf{a}_{c}$, then

Note that if $a\in\mathcal{J}_{V}$ satisfies $a>\mathbf{a}_{c}$ or $a=\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$, then there exist points $x_{1}(a)<x_{2}(a)<\dots<x_{r}(a)$ in $(c(a),\infty)$, for some $r\geq 2$, such that

On the other hand, if $V$ is a potential such that $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and $\mathbf{a}_{c}\in\mathcal{J}_{V}$, then there exist, for some $r\geq 1$, $x_{1}(\mathbf{a}_{c})<x_{2}(\mathbf{a}_{c})<\dots<x_{r}(\mathbf{a}_{c})$ in $(\mathbf{e},\infty)$ such that

The continuity of $x_{0}(a)$ for $a\not\in\mathcal{J}_{V}$ is a direct consequence of the continuity of $\mathbf{G}(x;a)$ in both $x$ and $a$. Let $\mathbf{a}_{c}\leq a_{1}<a_{2}$ and $a_{1},a_{2}\not\in\mathcal{J}_{V}$. If we assume $x_{0}(a_{1})\geq x_{0}(a_{2})$, then since $\mathbf{G}(x_{0}(a_{1});a_{1})>\mathbf{G}(x_{0}(a_{2});a_{1})$, we have

This is contradictory to the assumption that $x_{0}(a_{2})$ is the maximizer of $\mathbf{G}(x;a_{2})$. Thus $x_{0}(a_{1})<x_{0}(a_{2})$.

If $a_{0}\in\mathcal{J}_{V}$ and $a_{0}>\mathbf{a}_{c}$, then $\mathbf{G}(x;a_{0})$ attains its maximum in $[c(a_{0}),\infty)$ at $x_{1}(a_{0}),\dots,x_{r}(a_{0})$ for some $r\geq 2$ as in (39). It is easy to check from the continuity of $\mathbf{G}(x;a)$ in $a$ that $\lim_{a\uparrow a_{0}}x_{0}(a)=x_{1}(a_{0})$ and $\lim_{a\downarrow a_{0}}x_{0}(a)=x_{r}(a_{0})$. ∎

The secondary critical values for the spiked model are defined as the points $a\in\mathcal{J}_{V}$ such that $a>\mathbf{a}_{c}$.

For a potential $V$ such that $V(x)$ is convex for $x\geq\mathbf{e}$, $\mathcal{J}_{V}=\emptyset$ since $\mathbf{G}^{\prime}(x;a)$ is a decreasing function in $x\geq\mathbf{e}$. Hence there is no secondary critical value.

5 Statement of results: non-convex potentials

Let $V(x)$ be a potential satisfying the conditions (6)–(8). Let $I_{n}^{T}$ and $J_{n}^{T}(x_{*})$ be the intervals defined in (23) and (24), respectively.

For $a>\mathbf{a}_{c}$ such that $a\notin\mathcal{J}_{V}$, if $\mathbf{G}^{\prime\prime}(x_{0}(a))\neq 0$, then

When $a$ is at or near the critical value $\mathbf{a}_{c}$, we have the following result. The case when $a=\mathbf{a}_{c}$ is attained by setting $\alpha=0$.

Suppose that $V$ is a potential such that $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and $\mathbf{a}_{c}\notin\mathcal{J}_{V}$. Then for

Let $V$ be a potential such that $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$. If $\mathbf{a}_{c}\notin\mathcal{J}_{V}$ and $\mathbf{G}^{\prime\prime}(x_{0}(\mathbf{a}_{c});\mathbf{a}_{c})\neq 0$, then for

where the constant $p_{j,n}(\alpha)\in(0,1)$ is defined by (145). As a function of $\alpha$, $p_{j,n}(\alpha)$ is decreasing and satisfies $p_{j,n}(\alpha)\to 0$ as $\alpha\to\infty$ and $p_{j,n}(\alpha)\to 1$ as $\alpha\to-\infty$ for each fixed $n$. Also for each fixed $\alpha$, $p_{j,n}(\alpha)$ lies in a compact subset of $(0,1)$ for all large $n$.

When the potential $V(x)$ is convex for $x\geq\mathbf{e}$, then $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$, $\mathcal{J}_{V}=\emptyset$ (see Remark 1.2) and $\mathbf{G}^{\prime\prime}(x)<0$ for all $x>\mathbf{e}$. Hence Theorem 1.2 and Theorem 1.3(a) imply Theorem 1.1.

For $a$ at or near the secondary critical values $\mathcal{J}_{V}\setminus\{\mathbf{a}_{c}\}$, we have the following result.

Let $V$ be a potential such that $\mathcal{J}_{V}\neq\emptyset$. Let $a_{0}\in\mathcal{J}_{V}\setminus\{\mathbf{a}_{c}\}$ be a secondary critical point. If $\mathbf{G}(x;a_{0})$ attains its maximum $\mathbf{G}_{\max}(a_{0})$ at two points $x_{1}(a_{0})<x_{2}(a_{0})$ in $(c(a),\infty)$ and if $\mathbf{G}^{\prime\prime}(x_{1}(a_{0});a_{0})\neq 0$ and $\mathbf{G}^{\prime\prime}(x_{2}(a_{0});a_{0})\neq 0$, then for

where $p^{(1)}_{j,n}(\alpha)$ and $p^{(2)}_{j,n}(\alpha)\in(0,1)$ are defined in (164) and (158), and $p^{(1)}_{j,n}(\alpha)+p^{(2)}_{j,n}(\alpha)=1$. As a function of $\alpha$, $p^{(1)}_{j,n}(\alpha)$ is decreasing and satisfies $p^{(1)}_{j,n}(\alpha)\to 0$ as $\alpha\to\infty$ and $p^{(1)}_{j,n}(\alpha)\to 1$ as $\alpha\to-\infty$ for each fixed $n$. Also for each fixed $\alpha$, $p^{(1)}_{j,n}(\alpha)$ is in a compact subset of $(0,1)$ independent of $n$.

The above three theorems describe the ‘generic’ cases. The next part describes the three ‘exceptional cases’.

As the first exceptional case, suppose that in Theorem 1.4, the maximum of $\mathbf{G}$ is attained at more than two points. Let $x_{1}(a)<x_{2}(a)<\cdots<x_{r}(a)$ be these maximizers. If $\mathbf{G}^{\prime\prime}(x_{j}(a_{0});a_{0})\neq 0$ for all $k=1,\cdots,r$, then we have, for each $k=1,\cdots,r$,

for some $p_{j,n}^{(i)}(\alpha)\in(0,1)$ such that $p^{(1)}_{j,n}(\alpha)+\cdots+p^{(r)}_{j,n}(\alpha)=1$. Explicitly, $p^{(i)}_{j,n}(\alpha):=\frac{A_{i}(\alpha)}{A_{1}(\alpha)+\cdots+A_{r}(\alpha)}$ where $A_{i}(\alpha)$ is defined in (158). The situation when $\mathbf{a}_{c}\in\mathcal{J}_{V}$ in Theorem 1.3(b) is similar. In this case, the maximum of $\mathbf{G}(x;\mathbf{a}_{c})$ in $(c(\mathbf{a}_{c}),\infty)$ is attained at $x_{1}(\mathbf{a}_{c})<x_{2}(\mathbf{a}_{c})<\cdots<x_{r}(\mathbf{a}_{c})$ for some $r\geq 2$ (see (39)). Assume that $\mathbf{G}^{\prime\prime}(x_{i}(\mathbf{a}_{c});\mathbf{a}_{c})\neq 0$ for all $i=1,\cdots,r$. Then with $C_{i}(\alpha)$, $i=1,\cdots,r$, defined by (144) with $x_{0}(\mathbf{a}_{c})$ replaced by $x_{i}(\mathbf{a}_{c})$, set $p_{j,n}^{(i)}(\alpha):=\frac{C_{i}(\alpha)}{C_{0}+C_{1}(\alpha)+\cdots+C_{r}(\alpha)}$, $i=1,\cdots,r$, where $C_{0}$ is defined by (143). Then (47) holds with $p_{j,n}(\alpha)$ replaced by $p^{(0)}_{j,n}(\alpha):=1-p^{(1)}_{j,n}(\alpha)-\dots-p^{(r)}_{j,n}(\alpha)$. The limit in (48) is replaced by

The second exceptional case is when $\mathbf{a}_{c}\in\mathcal{J}_{V}$ in Theorem 1.3, case 1. This case is given in the following Theorem. In this case, there are two natural scalings in $a$.

Let $V$ be a potential such that $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$. Suppose that $\mathbf{a}_{c}\in\mathcal{J}_{V}$. Assume that $\mathbf{G}(x;\mathbf{a}_{c})$ attains its maximum at the unique point $x_{0}(\mathbf{a}_{c})\in(c(\mathbf{a}_{c}),\infty)$ and $\mathbf{G}^{\prime\prime}(x_{0}(\mathbf{a}_{c});\mathbf{a}_{c})\neq 0$. Then the following holds.

where $\alpha$ is in a compact subset of $(-\infty,0)$, we have

where $p_{j,n}(\alpha^{\prime})$ is defined in (280). As a function of $\alpha$, $p_{j,n}(\alpha^{\prime})$ is decreasing and satisfies $p_{j,n}(\alpha^{\prime})\to 0$ as $\alpha^{\prime}\to\infty$ and $p_{j,n}(\alpha^{\prime})\to 1$ as $\alpha^{\prime}\to-\infty$ for each fixed $n$. Also for each fixed $\alpha$, $p_{j,n}(\alpha)$ is in a compact subset of $(0,1)$ independent of $n$.

If the maximum of $\mathbf{G}(x;\mathbf{a}_{c})$ is attained at more than one point, then (58) should be changed in a natural way as in (53).

The third exceptional case is when the double derivative of $\mathbf{G}(x;a)$ vanishes at its maximizers. Then the function $G(x)$ is replaced by its higher analogue, and the scalings in the interval and $a$ are also changed accordingly. Concretely, in Theorem 1.2(b), if $\mathbf{G}^{\prime\prime}(x_{0}(a))=0$, then since $x_{0}(a)$ is the maximum point, there exists $k>1$ such that $\mathbf{G}^{(2k)}(x_{0}(a))<0$ and $\mathbf{G}^{(j)}(x_{0}(a))=0$ for all $j=1,2,\cdots,2k-1$. Then (43) is changed to

where the interval $\hat{J}_{n}^{T}(x_{*};k)$ is defined by

In Theorem 1.3(b), Theorem 1.4 and Theorem 1.5, $a$ is scaled as $a=a_{0}+\frac{\alpha}{n}$. When $\mathbf{G}^{\prime\prime}(x_{i}(a_{0});a_{0})=0$, ($i=0$ in Theorems 1.3(b) and 1.5, and $i=1,2$ in Theorem 1.4,) then this scaling also needs to be changed. For example, Theorem 1.4 is changed to the following theorem.

Let $a_{0}\in\mathcal{J}_{V}\setminus\{\mathbf{a}_{c}\}$ be a secondary critical value. Assume that $\mathbf{G}(x;a_{0})$ attains its maximum $\mathbf{G}_{\max}(a_{0})$ at two points $x_{1}(a_{0})<x_{2}(a_{0})$ in $(c(a),\infty)$. Suppose that $\mathbf{G}^{\prime\prime}(x_{1}(a_{0});a_{0})\neq 0$, and for some $k>1$, and suppose that $\mathbf{G}^{(2k)}(x_{2}(a_{0});a_{0})\neq 0$ and $\mathbf{G}^{(i)}(x_{2}(a_{0});a_{0})=0$ for all $i=1,\dots,2k-1$. Then for

where the interval $\hat{J}_{n}^{T}(x;k)$ is defined by (60), $p^{(1)}_{j,n}(\alpha)$ and $p^{(2)}_{j,n}(\alpha)$ are defined in (182), and $p^{(1)}_{j,n}(\alpha)+p^{(2)}_{j,n}(\alpha)=1$. As a function of $\alpha$, $p^{(1)}_{j,n}(\alpha)$ is decreasing and satisfies $p^{(1)}_{j,n}(\alpha)\to 0$ as $\alpha\to\infty$ and $p^{(1)}_{j,n}(\alpha)\to 1$ as $\alpha\to-\infty$ for each fixed $n$. Also for each fixed $\alpha$, $p^{(1)}_{j,n}(\alpha)$ is in a compact subset of $(0,1)$ independent of $n$.

The changes needed for Theorem 1.3(b), and Theorem 1.5 are analogous. Also it may happen that the two or more of the exceptional cases occur simultaneously. Then one needs simply combine the results together in a straightforward way, and we skip the details.

We also remark that one can obtain the convergence in probability 1 as in (3) from the above theorems together with the fact that all of the limiting distributions decay rapidly at the tails.

An explicit example of a potential such that $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$ can be constructed as follows. We use the potential defined in [18, Formula (4.14)] (we change the original notation $e$ into $\bar{e}$ here):

From results in [18, Section 4], it is known that $V_{\bar{e},\epsilon}$ is a regular potential with the support of the equilibrium measure given by $[-2+O(\epsilon),2+O(\epsilon)]$. For all $x>\mathbf{e}=2+O(\epsilon)$, (15) holds. However, at $x=\bar{e}$,

for some $E(\epsilon)$ satisfying $E(\epsilon)=O(\epsilon)$ and $E(\epsilon)>0$. Hence for any $a$, $\mathbf{G}(\bar{e};a)-\mathbf{H}(\bar{e};a)=O(\epsilon)$. Now there exists $a\in(0,\frac{1}{2}V^{\prime}(\mathbf{e}))$ such that $c(a)\in(\mathbf{e},\bar{e})$ since the minimizer $c(a)$ of $\mathbf{H}(x;a)$ is continuous in $a\in(0,\infty)$, decreases strictly in $a\in(0,\frac{1}{2}V^{\prime}(\mathbf{e}))$, $\lim_{a\downarrow 0}c(a)=+\infty$ and $c(\frac{1}{2}V^{\prime}(\mathbf{e}))=\mathbf{e}$ (see the sentence after Definition 1.1). Since $\mathbf{H}(c(a);a)<\mathbf{H}(\bar{e};a)$, we have $\mathbf{G}(\bar{e};a)>\mathbf{H}(c(a);a)$ if $\epsilon$ is small enough. Then $a\in\mathcal{A}_{V}$ and $\mathbf{a}_{c}<\frac{1}{2}V_{\bar{e},\epsilon}^{\prime}(\mathbf{e})$ for each fixed $\bar{e}>2$ if $\epsilon>0$ is small enough.

The paper is organized as follows. The outline of the proof of theorems is given in Section 2. The results on the orthonormal polynomials and the kernel $K_{n-j,n}$ are summarized in Section 6. The proofs of the theorems are given in Sections 3, 4 and 5. We consider three cases, $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$, $\mathbf{a}_{c}>\frac{1}{2}V^{\prime}(\mathbf{e})$ and $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ separately. Throughout this paper we only consider $a>0$. The $a<0$ case is discussed briefly in the end of Section 2.

We would like to thank Marco Bertola, Robbie Buckingham, Seung-Yeop Lee and Virgil Pierce for keeping us informed of the progress of their work. The work of Jinho Baik was supported in part by NSF grants DMS075709.

Outline of the proof

be the orthonormal polynomial of degree $j$ with respect to the weight $e^{-nV(x)}$. Here take $\gamma_{j}(n)>0$ to make $p_{j}$ unique. Set

be the Christoffel-Darboux kernel. Define the constant

We study two kinds of intervals $E=I_{n}^{T}$ and $E=J_{n}^{T}(x_{*})$, $x_{*}>\mathbf{e}$.

Then it follows that, since the operator norm of $(1-\chi_{I_{n}^{T}}K_{n-j,n}\chi_{I_{n}^{T}})^{-1}$ is uniformly bounded from Corollary 6.3 (c), and $\psi_{n-j}\to 0$ in $L^{2}(I_{n}^{T})$ from (332), that

Then since $(1-\chi_{[T,\infty)}\mathcal{K}_{n-j,n}\chi_{[T,\infty)})^{-1}\to(1-\chi_{[T,\infty)}K_{\operatorname{Airy}}\chi_{[T,\infty)})^{-1}$ in operator norm from (343) and $n^{-1/6}\psi_{n-j}(\mathbf{e}+\beta^{-1}n^{-2/3}\xi)-\mathcal{B}_{j,n}(\mathbf{e})\operatorname{Ai}(\xi)\to 0$ in $L^{2}([T,\infty))$ by Corollary 6.1(d), we find that if $u_{j,n}-u_{n}\to 0$ in $L^{2}([T,\infty))$, then

In Sections 3, 4 and 5 we only consider $a>0$. When $a<0$, the largest eigenvalue in the spiked source model defined by (21) has the same distribution as the negative value of the smallest eigenvalue of the spiked source model that is defined by the same formula but with the potential function $\hat{V}(x)=V(-x)$ and the external source matrix $-\mathbf{A}_{n-j+1}$. Since $\hat{V}(x)$ is regular as long as $V(x)$ is, and the non-zero eigenvalue of $−\mathbf{A}_{n-j+1}$ is positive, the analysis in this paper applies for that spiked source model. We need to keep track of the smallest eigenvalue in the new spiked source model, and it can be done in the same way that we analyze the largest one. It can be checked that the limiting distribution of the smallest eigenvalue is not affected by the positive external source eigenvalue $a$, corresponding to the $a<0$ case of Theorem 1.2(a). We skip any further remarks.

Note that $(C\varphi_{n-j})(z)e^{naz}\to 0$ exponentially as $\Re(z)\to-\infty$ since $a>0$. Therefore, we can deform the contour and obtain

where, with a constant $C_{\Gamma}>\frac{1}{2a}$,

The contours are oriented as indicated in Figure 5. Therefore we find

Let $\delta$ be given in Proposition 6.1. Let $\epsilon<\min\{c(a)-\mathbf{e},2\delta\}$ be a small enough positive constant, independent of $n$, such that all maximizers of $\mathbf{G}(x;a)$ in $[c,\infty)$ are in the interval $(\mathbf{e}+\epsilon,\infty)$. Recall the asymptotics of $(C\varphi_{n-j})(z)$ summarized in Section 6. Since the contours $\Gamma_{\pm}$ lie in $B_{\delta}$ (in Figure 10) in Section 6, from the asymptotic formula (320) for $(C\varphi_{n-j})(z)$,

We now use the method of steepest-descent to evaluate the integral asymptotically. By Lemma 1.1, $\mathbf{H}^{\prime}(c(a);a)=0$ and $\mathbf{H}^{\prime\prime}(c(a);a)>0$. It is straightforward to check, with the help of the formula of $\mathbf{g}(x)$ in (14), that for $z(t)=c+it$, $t>0$, the function $\Re\mathbf{H}(z(t);a)$ in $t$ satisfies

Also for $z(t)=c+iC_{\Gamma}-t\in\Gamma_{+}$, $t\geq 0$,

is negative for all $t\geq 0$ if $C_{\Gamma}>1/(2a)$. Hence $\Re\mathbf{H}(z;a)$ decreases as $z$ moves along $\Gamma_{+}$ counterclockwise. Similarly $\Re\mathbf{H}(z;a)$ increases as $z$ moves along $\Gamma_{-}$ counterclockwise. Therefore $\overline{\Gamma_{+}\cup\Gamma_{-}}$ is a curve of steep-descent for $\mathbf{H}$ with the saddle point at $z=c$. The fact that $z=c$ is a saddle point of $\mathbf{H}$ is the reason that we have split the integral in (80) at $c$.

Now consider the second integral in (85). From the asymptotics (318) for $\varphi_{n-j}$ in $B_{\delta}$,

Using that $M_{j,n}$ is uniformly bounded in any compact subset of $[\mathbf{e}+\epsilon,\infty)$, and $M_{j,n}(y)=O(y^{-j})$ as $y\to\infty$, and using that $\mathbf{G}(y;a)\to-\infty$ at least linearly, we obtain the trivial estimate that

where $\mathbf{G}_{\max}(a):=\max\{\mathbf{G}(y;a)\mid y\geq c(a)\}$. Together with (89), we obtain the following result. Recall the properties of $\mathbf{G}$ and $\mathbf{H}$ in Subsection 1.4.

Suppose that $a<\mathbf{a}_{c}$. Then $\mathbf{H}(c;a)>\mathbf{G}_{\max}(a)$. Therefore, (89) is exponentially larger than (91) and we obtain

Suppose that $\mathbf{a}_{c}<a<\frac{1}{2}V^{\prime}(\mathbf{e})$ (assuming that $V$ is such that $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$). Then $\mathbf{G}_{\max}(a)>\mathbf{H}(c;a)$ and hence (91) is exponentially larger than (89). Suppose that $a\notin\mathcal{J}_{V}$ and let $x_{0}=x_{0}(a)\in(c(a),\infty)$ be the unique point $\mathbf{G}_{\max}(a)$ is attained. If $\mathbf{G}^{\prime\prime}(x_{0};a)\neq 0$, using the Laplace’s method applied to (90) (using the properties of $M_{j,n}$ in Proposition 6.1 (a)), we obtain