Products of Independent Elliptic Random Matrices

Abstract

For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of Götze–Tikhomirov (On the asymptotic spectrum of products of independent random matrices, available at http://arxiv.org/abs/1012.2710) and O’Rourke–Soshnikov (J Probab 16(81):2219–2245, 2011) concerning the product of independent iid random matrices.

Appendix: Truncation

This section contains somewhat standard proofs of the truncation results in Sect. 6.1.

Proof of Lemma 6.1

We begin by observing that

$$\begin{aligned} {{\mathrm{Var}}}(\tilde{\xi }_i^{(N)}) = \mathbb {E}|\tilde{\xi }_i^{(N)}|^2 \le \mathbb {E}|\xi _i \mathbf {1}_{\{|\xi _i| \le N^\delta \}}|^2 \le 1 \end{aligned}$$

(8.1)

for $i=1,2$. We also have

$$\begin{aligned} \left| 1 - {{\mathrm{Var}}}(\tilde{\xi }_i^{(N)}) \right| \le 2 \mathbb {E}|\xi _i|^2 \mathbf {1}_{\{|\xi _i| > N^\delta \}} \le 2 \frac{M_{2+\tau }}{N^{\delta \tau }}, \end{aligned}$$

which verifies property (ii).

Thus, we take $N_0$ sufficiently large such that

$$\begin{aligned} {{\mathrm{Var}}}(\tilde{\xi }_i^{(N)}) \ge 1/2 \end{aligned}$$

(8.2)

for all $i=1,2$ and $N \ge N_0$. Then (i) follows by construction and the bound in (8.2).

It remains to prove (iii). Set $\tilde{\rho }^{(N)} := \mathbb {E}[ \tilde{\xi }_1^{(N)} \tilde{\xi }_2^{(N)}]$. Then by the Cauchy-Schwarz inequality and (8.1), we have

$$\begin{aligned} |\tilde{\rho }^{(N)} - \rho |&\le \sqrt{ \mathbb {E}|\xi _1|^2 \mathbf {1}_{\{|\xi _1| > N^\delta \}}} \sqrt{ \mathbb {E}|\xi _2|^2 \mathbf {1}_{\{|\xi _2| > N^\delta \}}} + \sum _{i=1}^2 \left( \sqrt{ \mathbb {E}|\xi _i|^2 \mathbf {1}_{\{|\xi _i| > N^\delta \}}} \right. \\&\quad \left. +\, \mathbb {E}|\xi _i|^2 \mathbf {1}_{\{|\xi _i| > N^\delta \}} \right) \le 5 \frac{M_{2+\tau }}{N^{\delta \tau /2}}. \end{aligned}$$

By property (ii) and (8.1), we obtain

$$\begin{aligned} |\hat{\rho }^{(N)} - \tilde{\rho }^{(N)}|&\le \mathbb {E}|\tilde{\xi }_1^{(N)} \tilde{\xi }_2^{(N)}| \left| \frac{1}{\sqrt{ {{\mathrm{Var}}}(\tilde{\xi }_1^{(N)}) {{\mathrm{Var}}}(\tilde{\xi }_2^{(N)})} } - 1 \right| \\&\le 2 \left| \sqrt{ {{\mathrm{Var}}}(\tilde{\xi }_1^{(N)}) {{\mathrm{Var}}}(\tilde{\xi }_2^{(N)})} - 1 \right| \\&\le 2 \left| {{\mathrm{Var}}}(\tilde{\xi }_1^{(N)}) {{\mathrm{Var}}}(\tilde{\xi }_2^{(N)}) - 1 \right| \\&\le 2 \left| {{\mathrm{Var}}}(\tilde{\xi }_1^{(N)}) - 1 \right| + 2 \left| {{\mathrm{Var}}}(\tilde{\xi }_2^{(N)}) - 1 \right| \\&\le 8 \frac{M_{2+\tau }}{N^{\delta \tau }}. \end{aligned}$$

Combining the bounds above completes the proof of property (iii). $\square$

Proof of Lemma 6.2

We begin with (6.2). By the block structure of $\mathbf {Y}_N$, it suffices to show that a.s.

$$\begin{aligned} \limsup _{N \rightarrow \infty } \frac{1}{N^2} \Vert \mathbf {Y}_{N,k} \Vert _2^2 < \infty \end{aligned}$$

(8.3)

for $k = 1,\ldots ,m$. We now decompose

$$\begin{aligned} \Vert \mathbf {Y}_{N,k} \Vert _2^2&= \sum _{1 \le i < j \le N} | (\mathbf {Y}_{N,k})_{ij} |^2 + \sum _{1 \le j < i \le N} | (\mathbf {Y}_{N,k})_{ij} |^2 + \sum _{i=1}^N | (\mathbf {Y}_{N,k})_{ii} |^2, \end{aligned}$$

where the summands in each sum are iid copies of $\xi _{k,1}, \xi _{k,2}$, and $\zeta _k$, respectively. Thus, applying the law of large numbers to each sum yields (8.3).

For (6.3), we decompose

$$\begin{aligned} \Vert \hat{\mathbf {Y}}_N \Vert _2^2&= \sum _{k=1}^m \Vert \hat{\mathbf {Y}}_{N,k} \Vert ^2_2 \\&\le 2 \sum _{k=1}^m \sum _{i,j=1}^N | (\tilde{\mathbf {Y}}_{N,k})_{ij} |^2 \\&\le 4 \sum _{k=1}^m \sum _{i,j=1}^N \left( |(\mathbf {Y}_{N,k})_{ij} |^2 + \mathbb {E}|(\mathbf {Y}_{N,k})_{ij} |^2\right) \\&\le 4 \left( \Vert \mathbf {Y}_N \Vert _2^2 + \mathbb {E}\Vert \mathbf {Y}_N \Vert _2^2 \right) \end{aligned}$$

by Lemma 6.1. As the atom variables have finite variance, (6.3) follows from (6.2).

It remains to prove (6.4). By the triangle inequality and the block structure of $\mathbf {Y}_N$ and $\hat{\mathbf {Y}}_N$, it suffices to show that almost surely, for $k=1,\ldots ,m$,

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{N^{\delta \tau }}{N^2} \Vert \mathbf {Y}_{N,k} - \tilde{\mathbf {Y}}_{N,k} \Vert _2^2 = 0 \quad \text {and} \quad \lim _{N \rightarrow \infty } \frac{N^{\delta \tau }}{N^2} \Vert \tilde{\mathbf {Y}}_N - \hat{\mathbf {Y}}_N \Vert _2^2 = 0. \end{aligned}$$

(8.4)

Fix $1 \le k \le m$. Then we have

$$\begin{aligned} \Vert \mathbf {Y}_{N,k} - \tilde{\mathbf {Y}}_{N,k} \Vert _2^2 \le 2 \sum _{i,j=1}^N \left( |(\mathbf {Y}_{N,k})_{ij}|^2 \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} + \mathbb {E}|(\mathbf {Y}_{N,k})_{ij}|^2 \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}}\right) . \end{aligned}$$

Since $\delta \tau < 1$, we have that a.s.

$$\begin{aligned} \limsup _{N \rightarrow \infty } \frac{N^{\delta \tau }}{N^2}&\sum _{i=1}^N \left( |(\mathbf {Y}_{N,k})_{ii}|^2 \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} + \mathbb {E}|(\mathbf {Y}_{N,k})_{ii}|^2 \right) \\&\le \limsup _{N \rightarrow \infty } \frac{N^{\delta \tau } }{N^2} \sum _{i=1}^N \left( |(\mathbf {Y}_{N,k})_{ii}|^2 + \mathbb {E}|(\mathbf {Y}_{N,k})_{ii}|^2 \right) = 0 \end{aligned}$$

by the law of large numbers. On the other hand,

$$\begin{aligned} \frac{N^{\delta \tau }}{N^2}&\sum _{1 \le i < j \le N} \left( |(\mathbf {Y}_{N,k})_{ij}|^2 \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} + \mathbb {E}|(\mathbf {Y}_{N,k})_{ij}|^2 \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} \right) \\&\le \frac{1}{N^2} \sum _{1 \le i < j \le N} \left( |(\mathbf {Y}_{N,k})_{ij}|^{2+\tau } \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} + \mathbb {E}|(\mathbf {Y}_{N,k})_{ij}|^{2+\tau } \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} \right) . \end{aligned}$$

By the dominated convergence theorem

$$\begin{aligned} \limsup _{N \rightarrow \infty } \frac{1}{N^2}&\mathbb {E}|(\mathbf {Y}_{N,k})_{ij}|^{2+\tau } \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} \le \limsup _{N \rightarrow \infty } \mathbb {E}|\xi _{k,1}|^{2+\tau } \mathbf {1}_{\{|\xi _{k,1}| > N^{\delta }\}} = 0. \end{aligned}$$

Furthermore, by the law of large numbers, we have a.s.

$$\begin{aligned} \limsup _{N \rightarrow \infty } \frac{1}{N^2} \sum _{1 \le i < j \le N} |(\mathbf {Y}_{N,k})_{ij}|^{2+\tau } \mathbf {1}_{\{|(\mathbf {Y}_{N,k})_{ij}| > N^\delta \}} = 0. \end{aligned}$$

The sum involving the indices $1 \le j < i \le N$ is handled similarly, and hence we conclude that a.s.

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{N^{\delta \tau }}{N^2} \Vert \mathbf {Y}_{N,k} - \tilde{\mathbf {Y}}_{N,k} \Vert _2^2 = 0. \end{aligned}$$

We now consider the second limit in (8.4). By Lemma 6.1, we obtain

$$\begin{aligned} \Vert \tilde{\mathbf {Y}}_{N} - \hat{\mathbf {Y}}_N \Vert _2^2&= \sum _{k=1}^m \sum _{i,j=1}^N |(\hat{\mathbf {Y}}_{N,k})_{ij}|^2 \left| \sqrt{ {{\mathrm{Var}}}( (\tilde{\mathbf {Y}}_{N,k})_{ij}) } - 1 \right| ^2 \\&\le \sum _{k=1}^m \sum _{i,j=1}^N |(\hat{\mathbf {Y}}_{N,k})_{ij}|^2 | {{\mathrm{Var}}}( (\tilde{\mathbf {Y}}_{N,k})_{ij}) - 1 |^2 \\&\le \frac{1}{N^{2 \delta \tau }} \sum _{k=1}^m \sum _{i,j=1}^N |(\hat{\mathbf {Y}}_{N,k})_{ij}|^2 \\&\le \frac{1}{N^{2 \delta \tau }} \Vert \hat{\mathbf {Y}}_{N} \Vert _2^2. \end{aligned}$$

Thus, by (6.3), a.s. we have

$$\begin{aligned} \limsup _{N \rightarrow \infty } \frac{N^{\delta \tau }}{N^2} \Vert \tilde{\mathbf {Y}}_{N} - \hat{\mathbf {Y}}_N \Vert _2^2 \le \limsup _{N \rightarrow \infty } \frac{1}{N^{2 + \delta \tau }} \Vert \hat{\mathbf {Y}}_N \Vert _2^2 = 0, \end{aligned}$$

and the proof of the lemma is complete. $\square$