Lectures 2 – 3 : Wigner ’ s semicircle law

As we set up last week, let M n = [X ij ] n i,j=1 be a symmetric n × n matrix with Random entries such that • X i.j = X j,i • X i,j s are iid for all i < j, and X jj are iid for all j with E[X 2 ij ] = 1, E[X i j] = 0 • All moments exists for each entries. We considered the eigenvector of this random matrix; λ 1 ≤ λ 2 ≤ · · · ≤ λ n which turns out to be random elements depending continuously on M n ; Lemma 1. If H n is a topological space of n × n matrix with topology derived from the usual metric on product Lebesgue measurable space, then λ i (H) is a continuous function on H n. Proof. Let H = [h ij ] n i,j=1 be an element in H n. We know that H k = k T r(H k)) = k λ k i So for example, H 2 = i λ 2 i Note that therefore H 2 ≥ max(λ n , −λ 1). Our goal is to obtain λ in terms of H. So it is good if we can say lim k→∞ H k → λ n because λ n dominates all the other eigen vectors, maybe except λ 1. Clearly, this logic might not work because of the presence of negative eigen values including λ 1. To fix this problem we may just shift the matrix by H. In particular, we can claim lim k→∞ k T r((H + HI) k) → λ n + H