On the concentration of eigenvalues of random symmetric matrices

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. Research supported in part by a USA — Israel BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research supported in part by a USA — Israel BSF grant and by a Bergmann Memorial Grant.