SPECTRA OF RANDOM LINEAR COMBINATIONS OF MATRICES DEFINED VIA REPRESENTATIONS AND COXETER GENERATORS OF THE SYMMETRIC GROUP

• 成果类型: Article
• 署名作者: Evans, Steven N.
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP418
• 发表日期: 2009
• 页码: 726-741
• 关键词: CENTRAL-LIMIT-THEOREM; plancherel measure; eigenvalues; characters

摘要

We consider the asymptotic behavior as n -> infinity of the spectra of random matrices of the form 1/root n-1 (n-1)Sigma(k=1) Z(nk)rho(n) ((k, k + 1)), where for each n the random variables Z(nk) are i.i.d. standard Gaussian and the matrices rho(n) ((k, k + 1)) are obtained by applying an irreducible unitary representation rho(n) of the symmetric group on {1, 2,...,n} to the transposition (k, k + 1) that interchanges k and k + 1 [thus, rho(n) ((k, k + 1)) is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on {1, 2,...,n} are indexed by partitions lambda(n) of n. A consequence of the results we establish is that if lambda(n),(1) >= lambda(n),(2) >= ... >= 0 is the partition of n corresponding to rho(n), mu(n),(1) >= mu(n),(2) >= ... >= 0 is the corresponding conjugate partition of n (i.e., the Young diagram of mu(n) is the transpose of the Young diagram of lambda(n)), lim(n ->infinity) lambda(n,j)/n = p(i) for each i >= 1, and lim(n ->infinity) mu(n,j)/n = q(j) for each j >= 1, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean theta Z and variance 1-theta(2), where theta is the constant Sigma(i)p(i)(2)-Sigma(j)q(j)(2) and Z is a standard Gaussian random variable.