Maxima of entries of Haar distributed matrices

Abstract

Let Γ n =(γ ij ) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also W n :=max1≤ i , j ≤ n |γ ij |. We obtain the limiting distribution and a strong limit theorem on W n . A tool has been developed to prove these results. It says that up to n/( log n)2 columns of Γ n can be approximated simultaneously by those of some Y n =(y ij ) in which y ij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n).