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),
35
http://dx.doi.org/10.1090/surv/171/02
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