On the concentration of eigenvalues of random symmetric matrices

Abstract

It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e − t 232 s2. The main ingredient in the proof is Talagrand's Inequality for concentration of measure in product spaces.