The entries of circular orthogonal ensembles

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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 = (vij(n))n×n, n ≥ 1}, as n→∞, n vij (n) converges in distribution to ℂN (0,1) for any i ≥ 1,j ≥ 1 with i ≤ j and n vii(n) converges in distribution to 2 · ℂN (0,1) for any i ≥ 1.

Original languageEnglish (US)
Article number063302
JournalJournal of Mathematical Physics
Issue number6
StatePublished - 2009

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