Order statistics of the moduli of the eigenvalues of product random matrices from polynomial ensembles

Let X1, . . . , XmN be independent random matrices of order N drawn from the polynomial ensembles of derivative type. For any fixed n, we consider the limiting distribution of the nth largest modulus of the eigenvalues of X = ∏mN k=1Xk as N → ∞ where mN/N converges to some constant τ ∈ [0,∞). In particular, we find that the limiting distributions of spectral radii behave like that of products of independent complex Ginibre matrices.


Introduction and main results
w(|z k | 2 ), (z 1 , . . . , z N ) ∈ C N , (1.1) where Z N is the normalisation constant. The function w : R + → R + is called the weight function of X and satisfies that w(e x ) is an Pólya frequency function of order N .
Let X 1 , X 2 be two independent random matrices drawn from the polynomial matrix ensembles of derivative type with weight function w 1 , w 2 respectively. Then X 1 X 2 is also drawn from such ensembles with weight w 1 * w 2 which is the Mellin convolution of w 1 and w 2 . Moreover, if we normalize w k to be the probability density functions of some independent nonnegative random variables, say ζ k , k = 1, 2, then the weight function * Harbin Institute of Technology, China. E-mail: yh _ wang@hit.edu.cn corresponding to X 1 X 2 is the probability density function of ζ 1 ζ 2 , see ref. [14] for details.
There are several known random matrices of this type, such as induced complex Ginibre matrices [7] (particular the complex Ginibre matrices [9]) and their products [1,2], the induced Jacobi ensembles (also known as the truncated unitary matrices [21]) and their products [1,4].
Rider [15,16] proved that the distributions of the appropriately shifted and rescaled spectral radii of complex Ginibre matrices converge to the Gumbel distribution, refer [17] for real Ginibre matrices. Later, Chafaï and Péché [6] showed that this phenomenon is also true for the random normal matrix with weight function w(r) = e −N V ( √ r) , r ∈ R + , (1.2) where V (r) is called potential function and possesses appropriate differential and convex conditions. On the other hand, if w(e r ) is a Pólya frequency function of order N , (1.1) also describes the eigenvalue distribution of a polynomial random matrix ensemble with derivative type. For instance, we may take w(r) = e −N r 2α , α > 0. Recently, Jiang and Qi [11] investigated the limiting behaviors of the spectral radii of truncated unitary matrices and products of independent complex Ginibre matrices. Also, the distribution of the smallest modulus was investigated [3,5].
In this paper, we are interested in the order statistics of the moduli of the eigenvalues of such product random matrices. Let {m N } ∞ N =1 be a sequence of positive integers.
Assume that the limit of m N /N exists as N tends to infinity and denote it by τ . For each N , let X m N be independent random matrices of order N drawn from the polynomial matrix ensembles of derivative type with weight functions respectively. Denote the eigenvalues of the product matrix X =  That is, for any fixed integer 1 ≤ n ≤ N , |z| (n) is the nth largest of {|z k | : k = 1, 2, . . . , N }. Particularly, |z| (1) is the spectral radius of X.
Our analysis is based on the following structural results of the moduli of the eigenvalues. Proposition 1.2. Let {ξ l,n : 1 ≤ l ≤ m N , 1 ≤ n ≤ N } be independent random variables. The probability density function of each ξ l,n is Then, we have the following identity in distribution (|z| (1) , . . . , |z| (N ) ) d = (ξ (1) , . . . , ξ (N ) ), (1.6) where ξ (1) ≥ . . . ≥ ξ (N ) is the order statistics of independent random variables ξ n = m N l=1 ξ l,n , n = 1, . . . , N . functions w k (e r ) given by (1.3) should be Pólya frequency functions of order N , the analysis in this article also works for the particle system (1.1) on complex plane C whose weight function is the Mellin convolution of w k , k = 1, 2, . . . , m N . Our analysis in this paper works for the potential functions V possess the following conditions (a)-(d), but does not need the weight function to be Pólya frequency functions. Thus, we do not require w k (e r )'s to be Pólya frequency functions except for considering polynomial matrix ensembles.
Hereinafter, we always suppose that potential functions V possess the following properties: (a) V is defined on R + and taken values in R∪{∞} with domain D := {t ∈ R + : V (t) < ∞}.
(b) V has continuous derivatives up to fourth order on its domain D.
(c) There is a unique minimum of F (t) := V (t)−2 log t and denote it by t (0) . Furthermore, t (0) is an interior point of D and for every ε > 0, automatically satisfied for N large enough. We would like to point out that there are random matrices which are drawn from the polynomial matrix ensembles with derivative type but do not satisfy some of the above conditions. For instance, the truncated unitary matrices in the weak non-unitary case and the spherical ensembles [11]. Consider random matrix A −1 B where A and B are independent complex Ginibre matrices of order N , that is A −1 B is a spherical ensemble of order N . The corresponding weight function is where V N (r) = (1 + N −1 ) log(1 + r 2 ). The potential function V N depends on N . If we treat w(r 2 ) = 1 1 + r 2 e −N V (r) , where V (r) = log(1 + r 2 ), then the potential function V does not satisfy condition (c).
We would like to mention that the potential function V does not need to be convex as in [6]. Technically, condition (c) makes it possible to apply Laplace method and thus the unique minimum point t (0) of F (t) is a stationary point.
By condition (c), we denote the unique minimum point of V l (t) − 2 log t by t  (1) If τ = 0, suppose that inf N ≥1 {a N } > 0. Furthermore, we always assume that m N ≤ N and N is large enough such that is positive. Then, for all x ∈ R, the following holds pointwise (2) If τ ∈ (0, ∞), and if there exist a 0 ∈ (0, ∞) and b 0 ∈ R such that lim N →∞ a N = a 0 and lim N →∞ b N = b 0 . Then, for all x ∈ R, the following holds pointwise Taking m N ≡ 1 and V (r) = r 2 , the limiting distribution of |z| (n) for any given natural number n was obtained by [15,16]. In particular, after taking m N ≡ 1, the limiting result of the spectral radii |z| (1) was obtained by [6] for general potential function V .
is the product of independent complex Ginibre matrices. The corresponding limiting distributions of the spectral radii of X were obtained by [11].
The strategies of the proof. With the notation in Proposition 1.2, set n=1 P N,n = 1 and N n=N −j N +1 P N,n converges to some distribution function. To analyze the asymptotics of the latter we show that {log ξ l,n : 1 ≤ l ≤ m N , N − j N < n ≤ N } can be replaced by a set of independent random normal variables with appropriate means and variances by Talagrand inequality, see Lemma 2.2 and Proposition 2.3 for details. Then, using the same arguments as in [15] for the Riemann sum approximation, we obtain the desired results.

Proof of Theorem 1.5
We would like to introduce some notation. Let For each l and n, introduce F l, Because we just interest in the large N limits of moduli of eigenvalues, in this article we always assume that N is large.
Next lemma provides some properties related to the potential function.
Denote the unique minimum point of F n by t n and setα n := t 2 n F n (t n ) andβ n := t 3 n F n (t n ). Similarly, we denote the unique minimum point of F by t 0 and setα := ). Now, we are going to prove (2.5). Set g(t) = tV (t). It is easy to see that g(t n ) = 2n−1 N .
Lemma 2.2. With the notation of Lemma 2.1, for each N − j N < n ≤ N , let ζ n be a positive random variable whose p.d.f. is given by Then, for N large enough, we have (I) The following asymptotics hold, (II) There exist a standard normal random variable η and a positive constant K depending on the derivatives of V up to fourth order, such that (II) It is easy to check that the probability density function of is given by (2.10) Taking change of variable Here ϕ(t) is the probability density function of standard normal random variable and G n (t) = V (t n t) − 2n−1 N log t. Applying the Laplace approximation, there exists some constant K > 0, depending on the derivatives of G n (t) up to fourth order, such that (2.13) By Talagrand inequality [18], one obtains where W 2 is the 2-nd Wasserstein distance. The Kantorovich duality [19, p.19] implies that there exists a standard normal random variable η such that The proof is completed.
Motivated by the Lemma 2.6 in [11], we have the following replacement principle.
η l,n α l,n N + E log ξ l,n ≤ log f N (x) . (2.16) Proof. Recall that f N (x) is defined by (2.2). When τ = 0, for large number N , we have P max To deduce the last equality, we have used that log(1 + y) = y + o(y) for y in a small neighborhood of 0. And for τ ∈ (0, ∞), we have To prove this proposition, by Slutsky's theorem, it is sufficient to prove that there is a set of independent normal random variables {η l,n : converges to 0 in probability. Actually, according to Lemma 2.2, there is a set of independent standard normal random variables such that (2.9) holds for some constant K > 0 uniformly. For any ε > 0, one has  Now, let N tend to infinity, we get the desired result.
We have the following simple but useful lemma.
Order statistics of the moduli of the eigenvalues of product ensembles are nonincreasing in n.
Proof. Introduce Then differentiate this quantity with respect to y, by the Gurland's inequality [10], we Let c > 2 be some constant, and set Here x denotes the largest integer less than or equal to x. Then, for any x ∈ R if τ = 0 and for x > 0 if τ ∈ (0, ∞).
Proof. Case I: τ = 0. Let {η l,N −j N : l = 1, . . . , m N } be a set of independent standard normal random variables such that (2.9) holds. The following string of inequalities hold  Recalling the definition of j N as in (2.27), also taking Lemma 2.1 and statement (I) in (2.29) Applying Lemma 2.6 bellow, and then combining (2.28) and (2.29), we obtain (2.30) According to Lemma 2.4, one has lim N →∞ Case II: τ ∈ (0, ∞). We have the following string of inequalities Lemma 2.6. For any random variables X, Y and constant x, y, the following inequality holds P(X + Y > x + y) ≤ P(X > x) + P(|Y | > y).

(2.32)
Proof. The desired result can be induced directly from the following string of relations, The following lemma will be used in the proof of our main theorem frequently.
Proof. Here, we just give a proof of the last assertion. Notice that For sup j c N j tends to 0, the big O item tends to 0. Thus the desired result is obtained.
Proof of Theorem 1.5. We split the proof into two steps.
Step 1 is to the study the limiting distribution of the spectral radius |z| (1) . Let j N be as in (2.27). Proposition 2.3 and Lemma 2.5 imply that there are independent standard normal random variables {η l,n : η l,n α l,n N + E log ξ l,n ≤ log f N (x) .
Recalling the definition of f N (x) given by (2.2), we obtain and η is a standard normal random variable. Lemma 2.1 is used again to deduce (2.39). If τ = 0, by a similar argument as in [15], applying the Riemann sum approximation, (2.43) According to (2.39) and Lemma 2.7, we have As a byproduct, we also have lim N →∞ (2.45) If τ ∈ (0, ∞), by Lebesgue dominated convergence theorem and Lemma 2.7 we obtain holds for every x > 0. Here Φ(x) is the distribution function of the standard normal random variable.
Step 2 is to analysis the limit of |z| (n) for any fixed positive integer n. The probability that ξ (n) less than or equal to f N (x) is the sum from k = 0 to n − 1 of the probabilities that there are exactly k of {ξ i : 1 ≤ i ≤ N } larger than f N (x). Thus, applying Proposition 1.2, we know that . (2.47) The prefactor in (2.47) is just the probability that the spectral radius is not more than f N (x) which can be found in the previous step. So we just need to investigate the limit of the summation in (2.47).

Concluding
In this paper, we have investigated the asymptotic behavior of the kth large modulus of the eigenvalues of products of m N independent complex random matrices drawn from the polynomial random matrix ensembles with derivative type where k is a fixed positive integer. It was shown that the limiting distribution of kth large modulus of the eigenvalues is the same as that of the single complex Ginibre matrix [16] for m N /N → 0 (particularly m N is a constant), while for m N /N → τ ∈ (0, ∞) the results are generalization of the results for the spectral radii of products of independent complex Ginibre matrices [11]. Our analysis suggests that the distribution of the kth large modulus is determined by j N (2.27) independent random variables. When m N /N → 0 the j N independent random variables are nearly identically distributed according to Gaussian distribution, so the limiting distribution of spectral radii is Gumbel distribution. When m N /N → τ ∈ (0, ∞) the j N random variables are just independent, thus the distribution of largest spectral modulli is the products of distributions of these random variables whenever the latter is indeed a distribution.