The Entries of Circular Orthogonal Ensembles

Let V = (vij)n×n be a circular orthogonal ensemble. In this paper, for 1 ≤ m ≤ o( √ n/ log n), we give a bound for the tail probability of max1≤i,j≤m |vij − (1/n)y′ iyj |, where Y = (y1, · · · ,yn) is a certain n×n matrix whose entries are independent and identically distributed random variables with the standard complex normal distribution CN(0, 1). In particular, this implies that, for a sequence of such matrices {Vn = (v ij )n×n, n ≥ 1}, as n → ∞, √ nv (n) ij converges in distribution to CN(0, 1) for any i ≥ 1, j ≥ 1 with i 6= j, and √nv(n) ii converges in distribution to √ 2 · CN(0, 1) for any i ≥ 1.