The entries of circular orthogonal ensembles

Let V= (v_ij)_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 max_{1≤ i,j ≤ m} v_ij - (1/n) y′_iy_j , where Y= (y₁, y_n) 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 {V_n = (v_ij⁽ⁿ⁾)_n×n, n ≥ 1}, as n→∞, n v_ij (n) converges in distribution to ℂN (0,1) for any i ≥ 1,j ≥ 1 with i ≤ j and n v_ii⁽ⁿ⁾ converges in distribution to 2 · ℂN (0,1) for any i ≥ 1.