Math 289A: Lecture 13

Let be a sequence of deterministic Hermitian matrices, and let be any enumeration of the eigenvalues of Consider the corresponding sequence of random Hermitian matrices defined by where is a random unitary matrix whose distribution in is Haar measure. We then have a corresponding triangular array of real random variables,

whose th row

consists of the diagonal elements of the random matrix In terms of our input data we have

This is superficially similar to the setup for the Central Limit Theorem, but it is different: are exchangeable but not independent (make sure you understand why). Thus our first objective is simply to understand the asymptotic distribution of the -matrix element of a uniformly random Hermitian matrix with spectrum

In this lecture I will use the angled brackets notation favored by physicists to denote expectation. I will also use the angled bracket with a subscript “” for cumulants (the subscript could also stand for connected). So for example the variance of is

Problem 13.1. Prove that Also show that for we have for any constant (this translation invariance is a general property of cumulants).

Our Optimized Leading Cumulants Formula says that for any we have

where

and

is times a convergent positive series.

What is nice about the Optimized Leading Cumulants Formula is that it almost immediately suggests the possibility of Gaussian limiting behavior: because of the factor in the formula, it looks like cumulants of degree should vanish in the limit, which is the Gaussian signature. To actually establish this we need to determine how the rest of the formula behaves as grows large.

This is not so difficult. We have the finite sum

and the number of terms in the sum has no dependence on . The quantity is as (make sure you understand why). This leaves the fraction

The denominator is literally just a power of . As for the numerator, we have the bound

where is the spectral radius of Thus, we have

We therefore have the cumulant bound

where depends only on

Theorem 13.1. If the input sequence of Hermitian matrices is such that for some constant and all and if exists, then the centered random variable converges to a centered Gaussian of variance

Proof: By definition the first cumulant of is zero for every and from translation invariance of cumulants and the bound established above we have that all cumulants of degree converge to zero. From the Optimized Leading Cumulants formula and the estimates above, the only term in the second cumulant is exactly Therefore, converges to a centered Gaussian random variable of variance as and the convergence holds in distribution by the Moment Method.

-QED