 
Summary: On the concentration of eigenvalues of random symmetric matrices
Noga Alon
Michael Krivelevich
Van H. Vu
February 22, 2002
Abstract
It is shown that for every 1 s n, the probability that the sth largest eigenvalue of a random
symmetric nbyn matrix with independent random entries of absolute value at most 1 deviates from
its median by more than t is at most 4et2
/32s2
. The main ingredient in the proof is Talagrand's
Inequality for concentration of measure in product spaces.
1 Introduction
In this short paper we consider the eigenvalues of random symmetric matrices whose diagonal and upper
diagonal entries are independent real random variables. Our goal is to study the concentration of the
largest eigenvalues. For a symmetric real nbyn matrix A, let 1(A) 2(A) . . . n(A) be its
eigenvalues.
There are numerous papers dealing with eigenvalues of random symmetric matrices. The most
celebrated result in this field is probably the so called Semicircle Law due to Wigner ([10], [11]) describing
the limiting behavior of the bulk of the spectrum of random symmetric matrices under certain regularity
