Distribution of Singular Values of Random Band Matrices; Marchenko–Pastur Law and More

Abstract

We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where X is an $n\times n$ band matrix of bandwidth $b_{n}$ and R is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko–Pastur law.

Appendix

In this section we list the results which were used in the Sect. 3.

Lemma 8.1

(Lemma 2.3, [21]) Let P, Q be two rectangular matrices of the same size. Then for any $x,y\ge 0$,

$$\begin{aligned} \mu _{(P+Q)(P+Q)^{*}}(x+y,\infty )\le \mu _{PP^{*}}(x,\infty )+\mu _{QQ^{*}}(y,\infty ). \end{aligned}$$

Lemma 8.2

(Sherman–Morrison formula) Let $P_{n\times n}$ and $(P+vv^{*})$ be invertible matrices, where $v\in \mathbb {C}^{n}$. Then we have

$$\begin{aligned} (P+vv^{*})^{-1}=P^{-1}-\frac{P^{-1}vv^{*}P^{-1}}{1+v^{*}P^{-1}v}. \end{aligned}$$

In particular,

$$\begin{aligned} v^{*}(P+vv^{*})^{-1}=\frac{v^{*}P^{-1}}{1+v^{*}P^{-1}v}. \end{aligned}$$

Lemma 8.3

(Lemma 2.6, [21]) Let P, Q be $n\times n$ matrices such that Q is Hermitian. Then for any $r\in \mathbb {C}^{n}$ and $z=E+i\eta \in \mathbb {C}^{+}$ we have

$$\begin{aligned} \left| \text {tr}\left( (Q-zI)^{-1}-(Q+rr^{*}-zI)^{-1}\right) P\right| =\left| \frac{r^{*}(Q-zI)^{-1}P(Q-zI)^{-1}r}{1+r^{*}(Q-zI)^{-1}r}\right| \le \frac{\Vert P\Vert }{\eta }. \end{aligned}$$

Lemma 8.4

([2], Lemma 1) Let $\{X_{n}\}_{n}$ be a sequence of random variables such that $|X_{n}|\le K_{n}$ almost surely, and $\mathbb {E}[X_{i_{1}}X_{i_{2}}\ldots X_{i_{k}}]=0$ for all $k\in \mathbb {N},\;i_{1}<i_{2}<\cdots <i_{k}$. Then for every $\lambda \in \mathbb {R}$ we have

$$\begin{aligned} \mathbb {E}\left[ \exp \left\{ \lambda \sum _{i=1}^{n}X_{i}\right\} \right] \le \exp \left\{ \frac{\lambda ^{2}}{2}\sum _{i=1}^{n}K_{i}^{2}\right\} . \end{aligned}$$

In particular, for any $t>0$ we have

$$\begin{aligned} \mathbb {P}\left( \left| \sum _{i=1}^{n}X_{i}\right| >t\right) \le 2\exp \left\{ -\frac{t^{2}}{2\sum _{i=1}^{n}K_{i}^{2}}\right\} . \end{aligned}$$

Lemma 8.5

Let P, Q be two $n\times n$ matrices, then

$$\begin{aligned} \Vert \mu _{PP^{*}}-\mu _{QQ^{*}}\Vert \le \frac{2}{n}rank(P-Q), \end{aligned}$$

where $\Vert \cdot \Vert$ denotes the total variation norm between probability measures.

Proof

By Cauchy’s interlacing property,

$$\begin{aligned} \Vert \mu _{PP^{*}}-\mu _{QQ^{*}}\Vert\le & {} \frac{1}{n}\text {rank}(PP^{*}-QQ^{*})\\\le & {} \frac{1}{n}\text {rank}((P-Q)P^{*})+\frac{1}{n}\text {rank}(Q(P-Q)^{*})\\\le & {} \frac{2}{n}\text {rank}(P-Q). \end{aligned}$$

$\square$

Lemma 8.6

([4, Lemma C.3]) Let P and Q be $n\times n$ Hermition matrices, and $I\subset \{1,2,\ldots , n\}$, then

$$\begin{aligned} \left| \sum _{k\in I}(P-zI)^{-1}_{kk}-\sum _{k\in I}(Q-zI)_{kk}^{-1}\right| \le \frac{2}{\mathfrak {I}(z)}\text {rank}(P-Q). \end{aligned}$$

Lemma 8.7

Let $C_{j}$ and $B_{j}$ be defined in (6), $r_{j}$ be the jth column of R, and $I_{j}\subset \{1,2,\ldots , n\}$ be same as (1), and $z\in \mathbb {C}^{+}$. Then

$$\begin{aligned}&\mathbb {P}\left( \left| \sum _{k\in I_{j}}(C_{j}^{-1})_{kk}-\mathbb {E}\sum _{k\in I_{j}}(C_{j}^{-1})_{kk}\right|>t\right) \le 2\exp \left\{ -\frac{\mathfrak {I}(z)^{2}t^{2}}{32n}\right\} \\&\mathbb {P}\left( \left| \sum _{k\in I_{j}}(C_{j}^{-1}B_{j}^{-1})_{kk}-\mathbb {E}\sum _{k\in I_{j}}(C_{j}^{-1}B_{j}^{-1})_{kk}\right|>t\right) \le 2\exp \left\{ -\frac{\mathfrak {I}(z)^{2}t^{2}}{32n}\right\} \\&\mathbb {P}\left( \left| \sum _{k\in I_{j}}(C_{j}^{-1}r_{j}r_{j}^{*}C_{j}^{-1*})_{kk}-\mathbb {E}\sum _{k\in I_{j}}(C_{j}^{-1}r_{j}r_{j}^{*}C_{j}^{-1*})_{kk}\right|>t\right) \le 2\exp \left\{ -\frac{\mathfrak {I}(z)^{2}t^{2}}{32n}\right\} \\&\mathbb {P}\left( \left| \sum _{k\in I_{j}}(C_{j}^{-1}B_{j}^{-1}r_{j}r_{j}^{*}B^{-1*}C_{j}^{-1*})_{kk}-\mathbb {E}\sum _{k\in I_{j}}(C_{j}^{-1}B_{j}^{-1}r_{j}r_{j}^{*}B^{-1*}C_{j}^{-1*})_{kk}\right| >t\right) \\&\qquad \le 2\exp \left\{ -\frac{\mathfrak {I}(z)^{2}t^{2}}{32n}\right\} . \end{aligned}$$

Proof

Let $\mathcal {F}_{l}=\sigma \{y_{1},\ldots ,y_{l}\}$ be the $\sigma$-algebra generated by the column vectors $y_{1},\ldots , y_{l}$. Then, we can write

$$\begin{aligned}&\sum _{k\in I_{j}}(C_{j}^{-1})_{kk}-\mathbb {E}\sum _{k\in I_{j}}(C_{j}^{-1})_{kk}=\sum _{l=1}^{n}\left[ \mathbb {E}\left\{ \left. \sum _{k\in I_{j}}(C_{j}^{-1})_{kk}\right| \mathcal {F}_{l}\right\} -\mathbb {E}\left\{ \left. \sum _{k\in I_{j}}(C_{j}^{-1})_{kk}\right| \mathcal {F}_{l-1}\right\} \right] . \end{aligned}$$

Notice that for any two matrices P, Q, we have $\text {rank}(PP^{*}-QQ^{*})\le 2\text {rank}(P-Q)$ (from Lemma 8.5). Therefore, using the Lemmas 8.6 and 8.4, we can conclude the result. The remaining three equations can also be proved in the same way. $\square$