CHAPTER 2

Gaussian Ensembles: Semicircle Law

We study here the existence and properties of the limiting Normalized Counting

Measure (1.1.18) of eigenvalues of Gaussian matrices, given by (1.1.1) – (1.1.3) and

more general matrices (2.2.1) for n-independent intervals, more precisely, quantities

(1.1.22) and (1.1.23) in the global regime. In particular, we are going to prove that

the Normalized Counting Measure of eigenvalues converges weakly with probability

1 to the nonrandom measure, known as the semicircle law or the Wigner law,

proposed by Wigner in 1951 and proved later by several methods. We essentially

follow [391, 394].

2.1. Technical Means

Definition 2.1.1. Let m be a nonnegative finite measure on R. The function

(2.1.1) f(z) =

m(dλ)

λ − z

,

defined for all nonreal z, z = 0, is called the Stieltjes transform of m.

Proposition 2.1.2. Let f be the Stieltjes transform of a finite nonnegative

measure m. Then:

(i)f is analytic in C \ R, and f(z) = f(z);

(ii) f(z) · z 0 for z = 0;

(iii) |f(z)| ≤ m(R)/|z|, in particular, limη→∞ η|f(iη)| ≤ ∞;

(iv) for any function f possessing the above properties there exists a nonnegative

finite measure m on R such that f is its Stieltjes transform and

(2.1.2) lim

η→∞

η|f(iη)| = m(R);

(v) if Δ is an interval of R whose edges are not atoms of the measure m, then

we have the Stieltjes-Perron inversion formula

(2.1.3) m(Δ) = lim

→0+

1

π

Δ

f(λ + i)dλ;

(vi) the above one-to-one correspondence between finite nonnegative measures

and their Stieltjes transforms is continuous if we use the uniform convergence of

analytic functions on a compact set of infinite cardinality of C \ R for Stieltjes

transforms and the vague convergence for measures (see Definition 1.1.7) in general

and the weak convergence of probability measures if the r.h.s. of ( 2.1.2) is 1;

(vii) if for some λ ∈ R there exists the nontangential limit from the upper

half-plane

f(λ + i0) := lim

z→λ+i0

f(z),

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http://dx.doi.org/10.1090/surv/171/02