On the largest eigenvalue of a Hermitian random matrix model with spiked external source II. Higher rank cases

Introduction

Here $Z_{n}$ is the normalization constant so that $\int_{\mathcal{H}_{n}}p_{n}(M)dM=1$ where $dM$ denotes the Lebesgue measure. The sequence of probability spaces $(\mathcal{H}_{n},p_{n})$, $n=1,2,\cdots$, is called a Hermitian matrix model with external source matrices $A_{n}$, $n=1,2,\cdots$, and potential $V$. Note that due to the unitary invariance of $dM$ and the presence of the trace in the exponent of (1), the density $p_{n}(M)$ depends only on the eigenvalues of $\mathbf{A}_{n}$. Hence we may assume without loss of generality that $\mathbf{A}_{n}$ is a diagonal matrix.

The main focus of this paper is the special case when

for all $n\geq\mathbf{m}$ with fixed nonzero $\mathbf{a}_{1},\cdots,\mathbf{a}_{\mathbf{m}}$. (We also consider the case when $\mathbf{a}_{j}$ depend on $n$. Indeed, in transitional cases we assume $\mathbf{a}_{j}$ varies in $n$ but converges to a fixed value as $n\to\infty$.) In this case, $(\mathcal{H}_{n},p_{n})$ is called a Hermitian matrix model with spiked external source (spiked model for short) of rank $\mathbf{m}$. The main interest of this paper is to study how the limiting location and the fluctuation of the top eigenvalue(s) of $M$ as $n\to\infty$ depend on the “external source eigenvalues” $\mathbf{a}_{j}$.

The case when $\mathbf{m}=1$ (rank one case) was studied in to which we refer the readers for the background and motivations of the spiked models. See also , , , , , , , , , , , , , , , , and .

The spiked model of an arbitrary fixed rank was studied in great detail for Gaussian potential in and also for the so-called complex Wishart spiked models in . Let us review the Gaussian case here. Let $\mathbf{e}$ denote the right end-point of the support of the equilibrium measure associated to the potential $V$. For the Gaussian potential $V(x)=\frac{1}{2}x^{2}$, $\mathbf{e}=2$. Let $\xi_{\max}(n)$ denote the largest eigenvalue of the random matrix of size $n$. Then there is a constant $\beta>0$ such that

On the other hand, if $\mathbf{a}_{1}=\cdots=\mathbf{a}_{\mathbf{m}}>\frac{1}{2}V^{\prime}(\mathbf{e})$, there is a constant $x_{0}(\mathbf{a}_{1})>\mathbf{e}$ and $\gamma(\mathbf{a}_{1})>0$ such that

This continuity of $\xi_{\max}$ does not necessarily hold for general potentials. Indeed, for the rank $1$ case it was shown in that $\xi_{\max}$ may be a discontinuous function of the (unique) external eigenvalue for certain potentials. (It was shown that $\xi_{\max}$ is continuous when $V(x)$ is convex in $x\geq\mathbf{e}$. A criterion when the discontinuity occurs is given in .) If $V$ is such a a potential, then $\xi_{\max}$ for the potentials $sV$ is also discontinuous for all $s$ close enough to $1$. An example of such $V$ can be constructed by considering a two-well potential with a deep well of the left and a shallow well on the right. Nevertheless there still is universality: it was shown in the rank $1$ case that the limiting distribution of $\xi_{\max}(n)$ at the continuous points of $\xi_{\max}$ is (generically) same as that of the Gaussian potential. (It though varies depending on whether the external eigenvalue is sub-critical, critical, and super-critical.) Even more, at a discontinuous point, the limiting distribution is something new but it is still (generically) independent of $V$. In this paper we show that similar universality also holds for the higher rank case.

While and the current paper were being written, a work on a similar subject was announced in the recent preprints by Bertola et al. and . The major difference of their work and ours is that we take $\mathbf{a}_{j}$’s to be all distinct and keep $\mathbf{m}$ fixed, while take $\mathbf{a}_{j}$ to be identical and let $\mathbf{m}\to\infty$ with $\mathbf{m}=o(n)$. Hence these two works complement each other. The methods are different and it seems that each has an unique advantage in handling the situations mentioned above; see Remark 1.3 below. See also Section 1.1 of for a further comparison.

The starting point of analysis in this paper is a simple algebraic relation between the higher rank case and the rank $1$ case. The gap probability, for example, can be written as a finite determinant built out of the gap probabilities of rank $1$ cases.

In order to state this algebraic relation, we slightly generalize the setting of the spiked model. Note that in the definition of the density (1), the factor $n$ in front of $\operatorname{Tr}(V(M)-\mathbf{A}_{n}M)$ equals the dimension of the matrices $M$ and $\mathbf{A}_{n}$. We may take the factor different from the dimension and consider the following p.d.f. :

where $\xi_{j}$, $j=1,\cdots,d$ denote the eigenvalues of $M$. It is well known that (see e.g. )

is the probability that there are no more than $j-1$ eigenvalues in $E$. When $E=(x,\infty)$, this is precisely the cumulative distribution function (c.d.f.) of the $j$th largest eigenvalue. When $\mathbf{A}_{d}=0$ we denote (7) by $\mathcal{E}_{d,n}(E;s)$, and also define

Let $p_{j}(x;n)$, $j=0,1,\cdots$, be the orthonormal polynomials with respect to the (varying) measure $e^{-nV(x)}dx$ and set $\psi_{j}(x;n):=p_{j}(x;n)e^{-\frac{n}{2}V(x)}$. Define

The following identity relates the higher rank case to the rank one cases.

When some $\mathbf{a}_{j}$ are identical, the above theorem still holds by using L’Hôpital’s rule. This follows from the smooth dependence of the quantities above in $\mathbf{a}_{j}$’s which can be proved directly. The explicit smooth dependence of multiple orthogonal polynomials on $\mathbf{a}_{j}$’s, which is essentially equivalent to the smooth dependence of quantities in (11), is shown in, for example, in similar situations. However, in the rest of the paper, we consider only the case when $\mathbf{a}_{j}$ are all distinct.

From the above theorem, the study of the limiting distribution of the eigenvalue of higher rank case may be reduced to a study of rank $1$ case, which was done in . However, for the interesting cases when $\mathbf{a}_{j}$’s converge to the same number in the limit, the numerator and denominator both tend to zero and thus we need to perform suitable row and column operations and extract the common decaying factors to make the ratio finite. This requires us to extend the asymptotic result of Baik and Wang to include the sub-leading terms of the asymptotics of $\bar{\mathcal{E}}_{n-j+1,n}(\mathbf{a}_{k};E;s)$. Nevertheless we requires only the existence of the asymptotic expansion but not the exact formulas, and hence most of the extension of the result of Baik and Wang is straightforward. The technical part is the row and column operations and to show that the ratio becomes finite after factoring out the common terms.

2 Assumptions on potential V𝑉V and some preliminary notations

In this section, we first state the precise conditions on $V$. Then we fix some notations and discuss a few important results of the rank $1$ case.

Assume that $V$ satisfies the following three conditions:

At the end of this section, we will discuss additional technical assumptions on $V$.

Here the regularity of $V$ is a condition defined in which we do not state explicitly here. We note only that this condition holds for “generic” $V$ and for such $V$, the density $\Psi(x)$ of the associate equilibrium measure (the limiting empirical measure when there is no external source) vanishes like a square-root at the edges of its support.

In the usual unitary ensembles (with no external source), the condition (13) is typically replaced by $\frac{V(x)}{\log(x^{2}+1)}\to+\infty$ as $|x|\to\infty$ . Here (13) is needed to ensure that the probability density (1) is well defined for all (spiked) $\mathbf{A}_{n}$.

With the above assumptions, the support $\Psi(x)$ consists of finitely many intervals:

for some $N\geq 0$. Note that we allow in this paper that $N$ can be larger than . We denote the right end-point of $J$ by

By the condition (14), $\beta$ is a nonzero positive number. It is also known that under the above assumptions (see and ) for the usual unitary ensemble with no external source,

where $F_{\operatorname{0}}$ is the Tracy–Widom distribution (see (34) for definition.)

We now recall a few notations and results from the analysis of rank one case . Let

for $a>0$. These functions play an important role in the analysis of rank one case. Observe that $\mathbf{G}(\mathbf{e};a)=\mathbf{H}(\mathbf{e};a)$ from (20). The function $\mathbf{H}(x;a)$ is convex in $x\in[\mathbf{e},\infty)$. Let $c(a)\in[\mathbf{e},\infty)$ be the point at which $\mathbf{H}(x;a)$ takes its minimum. It is easy to check that $c(a)=\mathbf{e}$ for $a\geq\frac{1}{2}V^{\prime}(\mathbf{e})$ and $c(a)>\mathbf{e}$ for $a<\frac{1}{2}V^{\prime}(\mathbf{e})$.

Now let $\mathbf{a}_{c}$ be the critical value associated to $V$ defined by

In general, $\mathbf{a}_{c}\in(0,\frac{1}{2}V^{\prime}(\mathbf{e})]$. If $V(x)$ is convex for $x\geq\mathbf{e}$, then $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$. The limiting location of the largest eigenvalue in the rank one case depends on whether $a<\mathbf{a}_{c}$ or $a>\mathbf{a}_{c}$. Here we denote it by $\xi(a)$ to indicate the dependence of on $a$.

This is a discrete set. If $V(x)$ is convex in $x\geq\mathbf{e}$, then $\mathcal{J}_{V}=\emptyset$. For $a>\mathbf{a}_{c}$ such that $a\notin\mathcal{J}_{V}$, let $x_{0}(a)$ denote the point in $[c(a),\infty)$ at which $\mathbf{G}(x;a)$ takes its maximum. For such $a$, it was shown that $x_{0}(a)$ is a continuous, strictly increasing function. Moreover, $\xi(a)$, the limiting location of the largest eigenvalue in the rank one case, equals $x_{0}(a)$ in this case. On the other hand, if $a>\mathbf{a}_{c}$ and $a\in\mathcal{J}_{V}$, then $\xi(a)$ is a discrete random variable whose values are the maximizers of $\max_{x\in(c(a),\infty)}G(x;a)$ (there are at least two of them). We call $a>\mathbf{a}_{c}$ such that $a\in\mathcal{J}_{V}$ the secondary critical values.

Sub-critical case: On the other hand, if $a<\mathbf{a}_{c}$, then $\xi(a)=\mathbf{e}$.

Critical case: At the critical case when $a=\mathbf{a}_{c}$, $\xi(a)$ depends on whether $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ or $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$. In both cases, let us assume that $\mathbf{a}_{c}\notin\mathcal{J}_{V}$. Then when $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$, $\xi(a)=\mathbf{e}$ as in the sub-critical case. But when $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$, $\xi(a)$ is a discrete random variable whose value is either $\mathbf{e}$ or the unique maximizer $x_{0}(\mathbf{a}_{c})$ of $\max_{x\in(c(a),\infty)}G(x;a)$ (which equals $\mathbf{H}(c(\mathbf{a}_{c});\mathbf{a}_{c})$ from the definition (22)).

For the rest of the paper, we assume that $V$ is a potential such that

Note that under this assumption, all of $\mathbf{G}^{\prime\prime}(x_{i}(a);a)$ ($i=0,1,2$) are negative. In , these excluded cases are referred as “exceptional cases”. However, to be precise, even though it is reasonable to imagine that nonexceptional cases are generic in the sense of Kuijlaars and McLaughlin , this was not established in . This issue will be considered somewhere else.

The above conditions are trivially satisfied if $V(x)$, $x\geq\mathbf{e}$, is convex since in this case $\mathcal{J}_{V}=\emptyset$. We note that if $V$ is such that $\mathcal{J}_{V}$ not empty, then it is easy to see that $\mathcal{J}_{sV}$ is also nonempty for real number $s$ close enough to $1$. Also it is easy to find an example of nonconvex potential $V$ such that $\mathcal{J}_{V}\neq\emptyset$ by considering a double-well potential. (See Remark 1.6 of .)

The analysis of this paper applies to the excluded cases without much change but we do not include them here for the sake of presentation.

We use the following notations for two intervals that appear frequently:

3 Statement of main results

We state the results under the “genericity assumptions” (24)- (28), in addition to the conditions (12)- (14) discussed in the last subsection. We group the asymptotic results into sub-critical, super-critical and critical cases.

The first result is on the sub-critical case when all external eigenvalues are smaller than the critical value $\mathbf{a}_{c}$. In this case the external source does not change the location and the limiting distribution of the top eigenvalues.

where $\chi_{E}$ denotes the projection on the set $E$. Then

is the Tracy–Widom $j$-th eigenvalue distribution. In particular,

3.2 Super-critical case

In this section we consider the super-critical case in which some of the external source eigenvalues are strictly larger than the critical value $\mathbf{a}_{c}$. In this case large external source eigenvalues do have an effect on the top eigenvalues. We consider three sub-cases. In the first two cases, we assume that $\mathbf{a}_{j}\notin\mathcal{J}_{V}$ for all $j$. The first among these is the case when $\mathbf{a}_{j}$ are separated by $O(1)$ distances. In the second case, the external source eigenvalues are asymptotically the same. The third case is the secondary critical case when $\mathbf{a}_{j}$ are all asymptotically equal to some $a\in\mathcal{J}_{V}\setminus\{\mathbf{a}_{c}\}$. From the discussion of Section 1.2, the last case does not occur if $V(x)$ is convex for $x\in[\mathbf{e},\infty)$.

be the c.d.f. of the standard normal distribution.

Let $\mathbf{a}_{1},\cdots,\mathbf{a}_{\mathbf{m}}$ be fixed positive and distinct numbers. Suppose that there is $p\in\{1,\cdots,\mathbf{m}\}$ such that

where $J_{n}^{T}(x_{*})$ is defined in (30). We also have, for $j=1,2,\cdots$,

where $F^{(j)}_{\operatorname{0}}(T)$ is the Tracy–Widom $j$-th eigenvalue distribution defined in (33).

Theorem 1.3 demonstrates that each of the external source eigenvalues which is greater than $\mathbf{a}_{c}$ pulls exactly one eigenvalue out of the support of the equilibrium measure. The limiting location of each pulled-off eigenvalue depends only on the corresponding external source eigenvalue. The fluctuation of each pulled-off is Gaussian. The rest of the eigenvalues are unaffected by the external source eigenvalues asymptotically.

We now consider the situation when the external source eigenvalues are asymptotically the same. A non-Gaussian fluctuation appears when they converge together in a particular fashion. Define, for distinct $\alpha_{1},\cdots,\alpha_{k}$,

in terms of the notation (7) when $V(x)=x^{2}/2$ in (5). Hence $G_{k}(T;\alpha_{1},\cdots,\alpha_{k};s)$ is an expectation that arises from the $k\times k$ Gaussian Unitary ensemble (GUE) with external source $\operatorname{diag}(\alpha_{1},\cdots,\alpha_{\mathbf{m}})$. As a special case,

is the c.d.f. of the $j$-th largest eigenvalue of the $k$-dimensional GUE. When $j=k=1$, this equals $G(T)$.

Let $a$ be a fixed number such that $a>\mathbf{a}_{c}$, $a\notin\mathcal{J}_{V}$ and $\mathbf{G}^{\prime\prime}(x_{0}(a))\neq 0$. Set

Hence in this case the $\mathbf{m}$ eigenvalues which are outside of the bulk converge to the same location $x_{0}(a)$. After a scaling, they fluctuate as the eigenvalues of $\mathbf{m}\times\mathbf{m}$ GUE matrix with external source $\operatorname{diag}(\alpha_{1},\cdots,\alpha_{\mathbf{m}})$.

We can also consider the general case that for $a_{1},a_{2},\cdots>\mathbf{a}_{c}$ where $a_{j}\notin\mathcal{J}_{V}$ and $\mathbf{G}^{\prime\prime}(x_{0}(a_{j}))\neq 0$, $p_{1}$ external source eigenvalues are close to $a_{1}$, $p_{2}$ external source eigenvalues are close to $a_{2}$, etc. Then for each $j$, $p_{j}$ eigenvalues converge to $x_{0}(a_{j})$ and they fluctuate like the eigenvalues of $p_{j}\times p_{j}$ GUE with certain external source. This can be obtained by combining the proofs of Theorem 1.3 and 1.4 but it is tedious. We omit the proof.

In Theorem 1.4, we can also prove the analogue of (39) and show that for each $j\geq 1$, the $(\mathbf{m}+j)$-th eigenvalue converges to $\mathbf{e}$ and its limiting distribution is the Tracy–Widom $j$-th eigenvalue distribution defined in (33). Similar remark also applies to Theorem 1.5 and to Theorem 1.7 below.

We now consider the situation when all external source eigenvalues are near or at a secondary critical value of $V$. In this case, we will state the result under the assumption that the support of the equilibrium measure of $V$ consists of one interval (i.e. $N=0$ in (15)). This assumption is made only for the ease of statement: see Remark 1.6 below how the result is changed if $N>0$.

Let $a\in\mathcal{J}_{V}$ and we consider the situation when $\mathbf{m}$ external source eigenvalues converge to $a$. Under the assumption (25), the top $\mathbf{m}$ eigenvalues converge to one of the two possible locations, which we denote by $x_{1}(a)<x_{2}(a)$. How many of the eigenvalues converge to each of them? It turned out that any number is possible and it depends on how fast the external source eigenvalues converges to $a$. There are $\mathbf{m}$ distinct scalings. To each scaling indexed by an $m\in\{1,\cdots,\mathbf{m}\}$ a number $\mathbf{p}_{m}\in(0,1)$ is associated such that either one of the following two happens: with probability $\mathbf{p}_{m}$, the top $m-1$ eigenvalues converge to $x_{2}(a)$ and the next top $\mathbf{m}-m+1$ eigenvalues to $x_{1}(a)$, or with probability $1-\mathbf{p}_{m}$, the top $m$ eigenvalues converge to $x_{2}(a)$ and the next top $\mathbf{m}-m$ eigenvalues to $x_{1}(a)$. In order to describe $\mathbf{p}_{m}$, we need some definitions.

We show in Proposition 7.1(b) that the determinant of $\mathfrak{P}^{(a,\mathbf{m}-j),(b,j)}$ is nonzero and $(-1)^{\mathbf{m}(\mathbf{m}-1)/2}$ times the determinant is positive if $a<b$. We also define, for $c\neq 0$ and distinct real numbers $\alpha_{1},\cdots,\alpha_{\mathbf{m}}$,

where $j=0,1,\cdots,\mathbf{m}$. Note that $(-1)^{\mathbf{m}(\mathbf{m}-1)/2}\det[\mathfrak{Q}_{(0,\mathbf{m}-j),(c,j)}(\alpha_{1},\cdots,\alpha_{\mathbf{m}})]>0$ if $c>0$ and $\alpha_{1}>\cdots>\alpha_{\mathbf{m}}$.

Assume that the support of the equilibrium measure associated to $V$ consists of one interval. Let $a$ be a secondary critical value (i.e. $a\in\mathcal{J}_{V}\setminus\{\mathbf{a}_{c}\}$ ) such that $\mathbf{G}(x;a)$, $x\in(c(a),\infty)$, attains its maximum value at two points $x_{1}(a)<x_{2}(a)$. Assume that $\mathbf{G}^{\prime\prime}(x_{1}(a);a)\neq 0$ and $\mathbf{G}^{\prime\prime}(x_{2}(a);a)\neq 0$. Fix $m\in\{1,2,\cdots,\mathbf{m}\}$, and set

Suppose that the external source eigenvalues are

Here $\mathbf{p}_{m}$ is a number in $(0,1)$ defined by

Observe that one more eigenvalue is pulled off from $x_{1}(a)$ to $x_{2}(a)$ as $m$ increases by $1$. The eigenvalues clustered near each of $x_{1}(a)$ or $x_{2}(a)$ fluctuate like the eigenvalues of a GUE matrix of dimension equal to the cluster size.

Note that $\mathbf{p}_{m}$ is well defined even for nondistinct $\alpha_{j}$ if we apply l’Hôpital’s rule to the right-hand side of (55). The theorem holds without the assumption of distinctness but we do not pursue it here. See Remark 1.3.

When the support of equilibrium measure associated to the potential $V$ consists of more than one interval (i.e. $N>0$), the above theorem still holds after one change: the probability $\mathbf{p}_{m}$ depends on $n$. In the formula (45), $\mathcal{M}_{j}$ needs to be changed to $\mathcal{M}_{j,n}$ in [8, Formula (311)], which is expressible in terms of a Riemann theta function and depends on $n$ quasi-periodically. Nevertheless, it can be shown that $\mathbf{p}_{m}$ lies in a compact subset of $(0,1)$ for all large enough $n$ from Proposition 7.1. This is enough to extend the proof of the above theorem from $N=0$ case to $N>0$ case.

3.3 Critical case

The final two theorems concern the critical case. In this case the limiting location of the top eigenvalue(s) is about to break off from $\mathbf{e}$. Recall that the critical value $\mathbf{a}_{c}$ which is determined by the potential $V$ satisfies $\mathbf{a}_{c}\leq\frac{1}{2}V^{\prime}(\mathbf{e})$. Depending on whether $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ or $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$, the break-off is continuous or discontinuous. When $V(x)$ is convex in $x\in[\mathbf{e},\infty)$, we always have $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and the break-off is continuous.

When $s=1$, the function $F_{1}(T;\alpha;1)$ was defined in [4, Definition 1.3] and $F_{k}(T;\alpha_{1},\cdots,\alpha_{k};1)$ was introduced in [2, Theorem 1.1]. They are known to be distribution functions and can be expressed in terms of Painlevé II equation and its Lax pair equations. It is also known that $F_{1}(T;0;1)$ is the square of the GOE Tracy–Widom distribution (see [4, Formula (24)]). (The function $F_{k}(T;\alpha_{1},\cdots,\alpha_{k};1)$ is shown to be the limiting distribution of the largest eigenvalue in the spiked model of rank $k$ at the critical case for the potentials $V(x)=((1+c)x-c\log x)\chi_{(0,\infty)}(x)$ and $V(x)=x^{2}/2$ in and , respectively. It is easy to check from the determinantal point process structure that

is the limiting distribution of the $j$-th largest eigenvalue in these potentials even though this was not discussed in .)

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

We now consider the case with potential $V$ such that $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$. As usual, we assume $\mathbf{a}_{c}\notin\mathcal{J}_{V}$. As in Theorem 1.5 above, we also assume, for the ease of statement, that the support of the equilibrium measure of $V$ consists of a single interval. An analogue of Remark 1.6 applies to the multiple interval case.

Also recall the matrix $\mathfrak{Q}_{(0,\mathbf{m}-j),(c,j)}(\alpha_{1},\cdots,\alpha_{\mathbf{m}})$ defined in (47)

Recall the point $x_{0}(\mathbf{a}_{c})$ which is well defined when $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$ from Section 1.2. Note that $x_{0}(\mathbf{a}_{c})>c(\mathbf{a}_{c})>\mathbf{e}$. The point $x_{0}(\mathbf{a}_{c})$ is unique when $\mathbf{a}_{c}\notin\mathcal{J}_{V}$.

Let $V$ be a potential such that $\mathbf{a}_{c}<\frac{1}{2}V^{\prime}(\mathbf{e})$. Assume that $\mathbf{a}_{c}\notin\mathcal{J}_{V}$ and $\mathbf{G}^{\prime\prime}(x_{0}(\mathbf{a}_{c});\mathbf{a}_{c})\neq 0$. Fix $m\in\{1,2,\cdots,\mathbf{m}\}$ and suppose that

for distinct $\alpha_{1}>\cdots>\alpha_{\mathbf{m}}$, where

Hence in this case, some eigenvalues are pulled off the edge of the equilibrium measure and fluctuate as the eigenvalues of Gaussian unitary ensemble. The largest eigenvalue which is not pulled off the edge of the equilibrium measure fluctuates as the Tracy–Widom distribution.

The proof of this theorem is very close to the proof of Theorem 1.5 and we skip it.

In this paper we state only limit theorems for individual top eigenvalues. The analysis of this paper can be modified to obtain the limit theorems of joint distribution of top eigenvalues. The result is what one would expect. We skip the detail.

4 Comments on the proofs and organization of the paper

We prove the asymptotic results in Section 1.3 based on Theorem 1.1 and extensions of the asymptotic results for the rank one case of . The latter are mostly routine and we do not give full details. In employing the above strategy to prove Theorems 1.2 and 1.3, we need to prove the limit of the determinant in the denominator of (11) is nonzero. This is done in Section 7. The proofs of Theorems 1.4, 1.5 and 1.7 are more complicated since the denominator of (11) converges to zero and hence we need to show that the numerator and the denominator have the same vanishing factors in their asymptotics. This requires careful linear algebraic manipulations. After factoring out the vanishing term, the denominator converges to a certain determinant which we show again nonzero in Section 7. The proof of Theorem 1.6 also follows this general strategy but we use a variation of Theorem 1.1 and the analysis is more involved. We skip the proof of Theorem 1.7 since it is very close to that of Theorem 1.5.

The rest of the paper is organized as follows. Theorems 1.2–1.6 are proved in the Sections 2–6, respectively. As mentioned in the above paragraph, in the proofs in these sections, we need to show that a certain determinant is nonzero. This is done in Section 7 in a unifying way. The algebraic theorem, Theorem 1.1, that relates the higher rank case to the rank $1$ case is proved in Section 8.

Proof of Theorem 1.2: sub-critical case

Recall that the sub-critical case is when

We also assume that $\mathbf{a}_{j}$’s are all positive, distinct and fixed. In order to use Theorem 1.1, we need the asymptotics of $\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k};n)$ and $\bar{\mathcal{E}}_{n-j+1,n}(\mathbf{a}_{k};I_{n}^{T};s)=\frac{\mathcal{E}_{n-j+1,n}(\mathbf{a}_{k};I_{n}^{T};s)}{\mathcal{E}_{n-j+1}(I_{n}^{T};s)}$.

From [8, Formula (92)] we have for all $a\in(0,\mathbf{a}_{c})$

For the rank $1$ case, the analogue of (74) is (see [8, Formula (73)] (Only the $s=1$ case is given in but the same proof works for general $s\neq 1$.))

For $E=I_{n}^{T}$, $1-s\chi_{E}K_{n-j,n}\chi_{E}$ is invertible for all $s$ close enough to $s=1$. (This is because $\chi_{E}K_{n-j,n}\chi_{E}$ converges to $\chi_{[T,\infty)}K_{\operatorname{Airy}}\chi_{[T,\infty)}$ in operator norm when $E=I_{n}^{T}$ and since $\chi_{[T,\infty)}K_{\operatorname{Airy}}\chi_{[T,\infty)}$ has its spectrum in $[0,1)$. This appears in several places in the subsequence sections and we do not repeat this remark.) When $E=I_{n}^{T}$ and $a\in(0,\mathbf{a}_{c})$, the asymptotics of (79) for $s=1$ was obtained in [8, Section 3.3] by analyzing each term on the right-hand side. It was shown that both inner products are $O(n^{1/3})$ (see [8, Formulas (128), (131) and (332)]). Hence from (75) we find that

Combining (76) and (80) we obtain, for $a\in(0,\mathbf{a}_{c})$,

Inserting (81), (72) and (76) into the formula (11), we obtain

Proof of Theorem 1.3: super-critical case 1, separated external sources eigenvalues

Recall that we assume that there exists $p\in\{1,\cdots,\mathbf{m}\}$ such that the positive, distinct and fixed numbers

We assume, without loss of generality, that $\mathbf{a}_{1}>\mathbf{a}_{2}>\cdots>\mathbf{a}_{p}$. Furthermore, we assume that $\mathbf{a}_{j}\notin\mathcal{J}_{V}$ and $\mathbf{G}^{\prime\prime}(x_{0}(\mathbf{a}_{j}))\neq 0$ for each $j=1,\cdots,p$, where $x_{0}(a)$ is defined in the paragraph between (23) and (24) in Section 1.2.

To use Theorem 1.1, we need the asymptotics of $\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k};n)$, $\mathcal{E}_{n-j+1,n}(\mathbf{a}_{k};E;s)$, and $\mathcal{E}_{n-j+1}(E;s)$ for $E=I_{n}^{T}$ and $E=J_{n}^{T}(x_{*})$ with $x_{*}>\mathbf{e}$.

By Baik and Wang [8, Formula (93)], we find that if $\mathbf{a}_{c}<a<\frac{1}{2}V^{\prime}(\mathbf{e})$ and $a\not\in\mathcal{J}_{V}$, then

and $\mathcal{M}_{j,n}(z)$ is a generalization of $\mathcal{M}_{j}(z)$ in (45) when the support of the equilibrium measure is multi-interval (i.e., $N>0$). This again can be expressed explicitly in terms of a Riemann theta function; see [8, Formula (311)]. Note that $\mathcal{M}_{j,n}(z)$ depends on $n$ but is uniformly bounded in $n$, together with its derivatives, in any compact subset in $z$. For $a>\frac{1}{2}V^{\prime}(\mathbf{e})$, we have the asymptotics [8, Formula (188)] of $\mathbf{\Gamma}_{n-j}(a;n)$ which is an intermediate step toward the formula (84). It is easy to further compute the formula [8, Formula (188)] asymptotically (using the asymptotics of $\varphi_{n-j}(y)$) and we find that (84) also holds for $a>\frac{1}{2}V^{\prime}(\mathbf{e})$. The same applies to the case when $a=\frac{1}{2}V^{\prime}(\mathbf{e})>\mathbf{a}_{c}$ from the remark in the first paragraph of Baik and Wang [8, Section 5]. In conclusion, the formula (84) is valid for all $a>\mathbf{a}_{c}$ and $a\not\in\mathcal{J}_{V}$.

We now evaluate $\mathcal{E}_{n-j+1,n}(a;J_{n}^{T}(x_{*});s)$ for $a=\mathbf{a}_{j}$. From the assumption (83), there are two cases. The first is when $a>\mathbf{a}_{c}$ and $a\notin\mathcal{J}_{V}$ and the second is when $a<\mathbf{a}_{c}$. The formula (79) is the starting point.

uniformly in $s$ close to $1$. The estimates [8, Formulas (137) and (139)] can be extended straightforwardly to the set $J_{n}^{T}(x_{*})$ for $x_{*}$ not equal to $x_{0}(a)$ but still in $(\mathbf{e},\infty)$. Hence we obtain

uniformly in $s$ close to $1$. This is what is expected from (87) by taking $T=\infty$ for the first case and taking $T=-\infty$ the second case. Recall that these asymptotics are for $a>\mathbf{a}_{c}$ such that $a\notin\mathcal{J}_{V}$. The asymptotics (87) and (88) apply to $\mathbf{a}_{1},\cdots,\mathbf{a}_{p}$.

On the other hand, for $a<\mathbf{a}_{c}$, an estimate similar to (80) implies that (note that $J_{n}^{T}(x_{*})\subset I_{n}^{T}$ for any $x_{*}>\mathbf{e}$, (80) and $F_{\operatorname{0}}(T;s)\to 1$ as $T\to\infty$)

for $s$ close to $1$. This asymptotics applies to $\mathbf{a}_{p+1},\cdots,\mathbf{a}_{\mathbf{m}}$.

Now inserting the asymptotics (72), (84), (86), (87), (88) and (89) into (11), we obtain, for each $k=1,\cdots,p$,

The matrix $\mathfrak{P}$ is the $\mathbf{m}\times\mathbf{m}$ matrix defined in (227) up to column changes and hence the reciprocal of $\det[\mathfrak{P}]$ is bounded uniformly in $n$ by Proposition 7.1(a). Since the entries of $\mathfrak{P}$ are bounded uniformly in $n$, we find that

Since the left-hand side of (92) is analytic in $s$, we obtain (38) by taking derivatives and using (8).

Hence by (76) we have for $s$ close to $1$

This asymptotics and (84) are for $a=\mathbf{a}_{1},\cdots,\mathbf{a}_{p}$.

For $a=\mathbf{a}_{p+1},\cdots,\mathbf{a}_{\mathbf{m}}$ which are all less than $\mathbf{a}_{c}$, we can use the asymptotics (72) and (81).

Inserting them into (11), we obtain for $s$ close to $1$

where $\mathfrak{P}$ is same as (91). We obtain (39) by taking derivatives on both sides of (96).

Proof of Theorem 1.4: super-critical case 2, clustered external source eigenvalues

Let $a$ be a fixed number such that $a>\mathbf{a}_{c}$ and $a\notin\mathcal{J}_{V}$. Recall the definition $x_{0}(a)$ given in the paragraph between (23) and (24). As usual we assume that $\mathbf{G}^{\prime\prime}(x_{0}(a))\neq 0$. Set

for fixed distinct $\alpha_{1},\cdots,\alpha_{\mathbf{m}}$. To prove Theorem 1.4, we first evaluate the denominator of (11) asymptotically and then the numerator when $E=J_{n}^{T}(x_{0}(a))$.

The asymptotics of $\mathbf{\Gamma}_{n-j}(a)$ were evaluated in [8, Formula (93)] when $a$ is a constant. It is easy to see from the proof that [8, Formula (93)] is indeed uniform for $a$ in a compact subset of $(\mathbf{a}_{c},\infty)$ which is especially applicable when $\mathbf{a}_{k}$ given by (97). It is clear that the leading-order asymptotics of $\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k})$ are same for all $k=1,\cdots,\mathbf{m}$. This implies that the determinant $\det[\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k})]_{j,k=1}^{\mathbf{m}}$ converges to zero. Therefore, we need to evaluate the sub-leading terms in the asymptotics of $\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k})$ in order to determine the asymptotics of $\det[\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k})]_{j,k=1}^{\mathbf{m}}$.

for some $\delta>0$. Here $\mathbf{G}(y;a)$ is the function defined in (21) and $M_{j,n}(z)$ is an analytic function in a neighborhood of $z=x_{0}(a)$.

uniformly for $z$ in a neighborhood of $x_{0}(a)$, where $M_{j,n}^{(i-1)}$ is the $(i-1)$th derivative. As $n\to\infty$, $M_{j,n}(z)=\mathcal{M}_{j,n}(z)(1+O(n^{-1}))$ uniformly in $z$ in the same neighborhood for another analytic function $\mathcal{M}_{j,n}(z)$ which depends on quasi-periodically in $n$. (See [8, (319)]. This $\mathcal{M}_{j,n}(z)$ is the same $\mathcal{M}_{j,n}(z)$ appearing in (84).) A key property for our purpose is that a certain determinant involving $\mathcal{M}_{j,n}$ and its derivatives is nonzero, which is proved in Proposition 7.1(b) later. This is used when we consider $\det[\hat{\mathcal{P}}]$ and $\det[\mathcal{P}]$ below. Note that

since both functions are analytic. Each of $M_{j,n}^{(i-1)}(x_{0}(a))$ and $\mathcal{M}_{j,n}^{(i-1)}(x_{0}(a))$ are $O(1)$.

From the definition of $\mathbf{G}$ and (97),

Denote the $\mathbf{m}\times\mathbf{m}$ matrices

Note that all entries of these matrices are $O(1)$. We also set $\mathcal{N}$ to be an $\mathbf{m}\times\mathbf{m}$ diagonal matrix with entries

We now replace $\hat{\mathcal{P}}$ and $\hat{\mathcal{Q}}$ by matrices with entries given by the leading terms given in (100) and (105). Define the $\mathbf{m}\times\mathbf{m}$ matrix

The entries of this matrix are the leading term of the entries of the matrix $\hat{\mathcal{P}}$ (see (100)). From the general result Proposition 7.1(b) and by noting that $\mathcal{P}=\mathfrak{P}^{(x_{0}(a),\mathbf{m})}$ in the notation of Section 7, we find that $\det[\mathcal{P}]$ is nonzero. Moreover, $1/\det[\mathcal{P}]$, which depends on $n$, is uniformly bounded. This nonvanishing property is easy to check directly using the formula (45) when $N=0$ but is complicated when $N>0$. Now as $\hat{\mathcal{P}}=\mathcal{P}+o(1)$, we find that all entries of $\hat{\mathcal{P}}^{-1}$ are $O(1)$. Hence noting that the explicit dependence on $n$ of $\mathcal{N}$, we find that all entries of $n^{-\frac{\mathbf{m}+1}{2}}\hat{\mathcal{P}}\mathcal{N}^{-1}\hat{\mathcal{P}}^{-1}\mathcal{R}$ are $O(n^{-1/2})$.

Then $\hat{\mathcal{Q}}=\mathcal{Q}+o(1)$. Therefore, we find from (109) that

and this is nonzero when all $\alpha_{k}$’s are distinct.

We focus on the new determinant. As in the previous subsection, the determinant converges to zero and hence we need to find the leading asymptotics.

for some $\delta^{\prime}>0$ uniformly for $s$ close to $1$.

Now we evaluate the remaining inner product in (116). We actually evaluate the inner product multiplied by $\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k};n)$, which is what we need in view of (115). The leading-order asymptotics of this quantity was evaluated in [8, Section 3.4]. Here we need the sub-leading terms and this follows from a simple extension of the analysis for the leading term as follows. First, for all $x\in J_{n}^{T}(x_{0}(a))$, we have from [8, Formulas (106) and (330)] that

for some $\delta^{\prime\prime}>0$. This is same as [8, Formula (136)] (after substituting the asymptotics of $\mathbf{\Gamma}_{n-j}$) where the error term is written only as $o(1)$ instead of an exponentially small term. Recalling that $\mathbf{G}(x;a)$, $x\in(\mathbf{e},\infty)$, takes its unique maximum at $x=a$ by the assumption $a>\mathbf{a}_{c}$ and $a\notin\mathcal{J}_{V}$, and noting that $\mathbf{G}(x;\mathbf{a}_{k})$ is close to $\mathbf{G}(x;a)$ (see (104)), we find that for any $\epsilon>0$

for some $\delta^{\prime\prime\prime}>0$ where $E_{T,\epsilon}(x_{0}(a))$ is the interval

We now find from (116), (98), and (118) that

where $\mathcal{Q}_{T;s}$ is the $\mathbf{m}\times\mathbf{m}$ matrix with entries

Combining (112), (114), (121) and Theorem 1.1, we find that (recall (40) for the definition of $G_{k}$)

Proof of Theorem 1.5: secondary critical case

We assume that the support of the equilibrium measure associated to $V$ consists of one interval. Let $a\in\mathcal{J}_{V}\setminus\{\mathbf{a}_{c}\}$. Then $\mathbf{G}(x;a)$ attains its maximum in $(c(a),\infty)$ at more than one point. We assume that the maximum is achieved at two points, which we denote by $x_{1}(a)<x_{2}(a)$. We write $x_{1}(a)$ as $x_{1}$ and $x_{2}(a)$ as $x_{2}$ for notational convenience if there is no confusion. Set

We assume, as usual, that $\mathbf{G}^{\prime\prime}(x_{1};a)\neq 0$ and $\mathbf{G}^{\prime\prime}(x_{2};a)\neq 0$.

Throughout this section, we fix $m\in\{1,2,\cdots,\mathbf{m}\}$. Recall the definitions

for fixed distinct $\alpha_{1}>\cdots>\alpha_{\mathbf{m}}$.

The goal here is to prove the asymptotic formula (156) given at the end of this subsection.

Analysis in this section is similar to that of Section 4.1 but with the change that the main contribution to the integral formula of $\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k};n)$ comes from two intervals (near $x_{1}(a)$ and $x_{2}(a)$) instead of one interval as in (98). This is because of (124). We have a small enough $\epsilon>0$ and a corresponding $\delta>0$ such that

Since the two integrals are asymptotically of same order, the evaluation of the determinant \det\big{[}\mathbf{\Gamma}_{n-j}(\mathbf{a}_{k};n)\big{]}_{j,k=1}^{\mathbf{m}} is more complicated.

Using the Andréief’s formula in random matrix theory (see e.g. ), we have

For each variable $y_{k}$, the integral in $y_{k}$ is over $E_{1}\cup E_{2}$. Using the symmetry of the integrand in $y_{k}$ in the last line of (130) is symmetric in $y_{k}$, (130) equals

This is the partition function of the $j$-dimensional GUE. We have the following lemma.

Since $\mathbf{G}(y;\mathbf{a}_{j})-\mathbf{G}(x_{1};\mathbf{a}_{j})=\mathbf{G}(y;a^{\prime})-\mathbf{G}(x_{1};a^{\prime})+\frac{\alpha_{j}}{n}(y-x_{1})$ and $\mathbf{G}(x_{1};a)=\mathbf{G}(x_{2};a)$, (132) equals

and $\mathbf{E}^{(1)}$ is a matrix with entries satisfying, for each $j=1,\cdots,\mathbf{m}$,

Also for each $j=1,\cdots,\mathbf{m}$, (143) implies that

We consider the interval $J_{n}^{T}(x_{2})$ first. From (114) in Section 4.2,

Now (120) is changed to, as it happened to (128),

for some $\epsilon>0$ and $\delta>0$, where $E_{i}$ are in (129) and $E_{T,\epsilon}(x_{2})$ is the interval

Now the analysis of Section 5.1 goes through with the change that the measure $dy$ is changed to $(1-s\chi_{E_{T,\epsilon}(x_{2})}(y))dy$ for $y\in E_{2}$. Thus we find (cf. (131) and (132))

The analysis that yields Lemma 5.1 applies with trivial modifications and we obtain (cf. (134))

For the interval $J_{n}^{T}(x_{1})$, (158) is changed to

Corresponding to (131) and (160), we have

Proof of Theorem 1.6: critical case 1, continuous transition

We assume that the critical value $\mathbf{a}_{c}=\frac{1}{2}V^{\prime}(\mathbf{e})$ and suppose that $\mathbf{a}_{c}\notin\mathcal{J}_{V}$. Let

for fixed, distinct real numbers $\alpha_{1},\cdots,\alpha_{\mathbf{m}}$. Here $\beta$ is a positive constant defined in (17). The proof of this critical case is more involved than other cases. We first need to perform some algebraic manipulations of the determinant in Theorem 1.1 to make it asymptotically easy to evaluate.

We start with a formula that is equivalent to but slightly different from Theorem 1.1. From Lemma 8.1, which is an intermediate step toward the proof of Theorem 1.1,

and the column vector $\mathbf{w}(x):=(w_{1}(x),\cdots,w_{\mathbf{m}}(x))^{t}$ is given by

Using the notations above, we have (see (10))

By taking a linear combination of the last three rows and using the three-term recurrence relation (176), we can replace the last row in the above matrix by the vector

We then can replace the $(\mathbf{m}-1)$-th row similarly by using the two rows above. By repeating this process up to the third row, we obtain

Now we can change the last row of this new matrix to

without changing the determinant, by using a linear combination of the last three rows and the three-term recurrence relation again. We repeat this process up to the fifth row and obtain

We repeat the process and obtain, for even $\mathbf{m}$,

where $B_{j,n}(\mathbf{e})$ is the value of $B_{j,n}(z)$ at $z=\mathbf{e}$, and the function $B_{j,n}(z)$ is a function defined in [8, Proposition 6.1(b)] which appears in the asymptotic of orthonormal polynomial near the edge $\mathbf{e}$ of the support of the equilibrium measure. From the asymptotics of $B_{j,n}(z)$ [8, Formulas (323) and (313)], it was shown that $B_{j,n}(z)$ and its reciprocal are uniformly bounded in a neighborhood of $z=\mathbf{e}$.

When $\mathbf{m}$ is odd, we need to add an extra row $\langle(x-\mathbf{e})^{[\mathbf{m}/2]}\psi_{n-[\mathbf{m}/2]-1}(x),\mathbf{v}^{t}\rangle$ to the matrix to the right-hand side of (183) and the extra term $b_{n-[\mathbf{m}/2]-1}^{[\mathbf{m}/2]}$ needs to be multiplied on the left-hand side. In the remaining part of this section, we consider only even $\mathbf{m}$ since the odd $\mathbf{m}$ case can be solved by the same method.

(The result of involves $\mathcal{B}_{j,n}(\mathbf{e})$ in place of $B_{j,n}(\mathbf{e})$. But as $B_{j,n}(z)=\mathcal{B}_{j,n}(z)(1+O(n^{-1}))$ from [8, Formula (323)] and $1/\mathcal{B}_{j,n}(\mathbf{e})$ is uniformly bounded, the above statement follows.) Now observe that (see [8, Formula (223)])

By taking the derivatives of this identity with respect to $\alpha$, we find that

This can be written as the sum of the integrals (188) with the terms $(C\varphi_{n-j})(z)$ and $\varphi_{n-j}(z)$ replaced by $\varphi_{n-j-1}(z)-\frac{B_{j+1,n}(\mathbf{e})}{B_{j,n}(\mathbf{e})}\varphi_{n-j}(z)$ and $\varphi_{n-j-1}(z)-\frac{B_{j+1,n}(\mathbf{e})}{B_{j,n}(\mathbf{e})}\varphi_{n-j}(z)$, respectively. Then again the main contribution to the integrals come near $z=\mathbf{e}$. The precise behaviors of the integrands near $z$ are well known (see [8, Formula (322)]). First,

We note that from the explicit asymptotics [8, Formula (303), (304)] of $\gamma_{n-j}$, $\kappa_{j,n}$ and its reciprocal are bounded uniformly in $n$.

From this we can find the asymptotics of (194) in a similar form as (189). The resulting formula contains two integrals, one involving $\operatorname{Ai}$ and the other $\operatorname{Ai}^{\prime}$, since each of (195), (196), and (197) contains such two terms. Now notice that due to (199) and the change of variables $\xi=(z-\mathbf{e})\beta n^{2/3}$, the integral involving $\operatorname{Ai}$ is smaller than the integral involving $\operatorname{Ai}^{\prime}$ by the factor $O(n^{-1/3})$. Thus we find that

The integral above can be simplified by the identity

This identity is obtained by taking derivatives with respect to $\alpha$ of the identity

which follows from (191) after integrating by parts. Hence we obtain

Note that the $(j,k)$ entry of the determinant on the right-hand side of (205) is of the form $P_{j}(\alpha_{k})e^{\alpha_{k}^{3}/3}$ for some polynomial $P_{j}(x)$ of degree $j-1$ with leading coefficient $1$, (i.e., $P_{j}(x)=x^{j-1}+\cdots$), which are defined by the conditions

Therefore, by elementary row operations we find that the determinant is same as the determinant of the matrix $(\alpha_{k}^{j-1}e^{\alpha_{k}^{3}/3})_{j,k=1}^{\mathbf{m}}$. The determinant of this matrix is $\prod_{1\leq j<k\leq\mathbf{m}}(\alpha_{k}-\alpha_{j})\prod_{k=1}^{\mathbf{m}}e^{\alpha_{k}^{3}/3}$ and this is nonzero. Therefore, when $\mathbf{m}$ is even,

We have a similar result when $\mathbf{m}$ is odd.

We now evaluate the numerator of (173) when $E=I_{n}^{T}$. Note that $\psi_{n-j}$ is the only term that depends on $j$. Hence by using the same row operations as in Section 6.1 that lead to (183) and (185), we find that, when $\mathbf{m}$ is even,

where $E_{T,\epsilon}=I_{n}^{T}\setminus(\mathbf{e}+\epsilon,\infty)$ with a small enough constant $\epsilon$ and

is defined in (56). Using the asymptotics [8, Formula (197)] of $\Gamma_{n}(\mathbf{a}_{k})$ (It was given in terms of $\mathcal{B}_{j,n}(\mathbf{e})$ but we can change it to $B_{j,n}(\mathbf{e})$. See text in parenthesis below equation (190).), this implies that

Similarly, as in the argument for the asymptotics (201), we find that

Recall the polynomials $P_{j}(\alpha)$ defined in (206). We claim that

To see this, note that successive integrations by parts of the integral representation of the Airy function $\operatorname{Ai}(\xi)=\frac{1}{2\pi\sqrt{-1}}\int_{\infty e^{-\pi i/3}}^{\infty e^{\pi i/3}}e^{-\xi s+\frac{1}{3}s^{3}}ds$ imply that, for any $i$,

which proves the first identity of (216). Similarly, for any $i$,

Simple row operations then imply that the last determinant, without $o(1)$ term, equals

3 Proof of Theorem 1.6

From (173), (207), and (221), we find that

When $s=1$, $D_{\mathbf{m}}(s)$ is precisely the matrix $M$ defined in [2, Formula (3.36)] with $w_{k}=-\alpha_{k}$ (see [2, Formula (3.9)] for the definition of $E_{w}$ and [2, Formulas (3.4) and (1.10)] for the definition of $T_{1}$). A different formula of $\det(M)$ was then obtained [2, Formula (3.46)] in terms of function $f(x;w)$. Comparing with the case of $k=1$ of [2, Formula (1.16)], this function $f(x,w)=\frac{F_{1}(x;w)}{F_{0}(x)}$ and this implies that

From the definition of $F_{k}$ (58), we obtain (61).