Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group

Abstract

We consider the asymptotic behavior as $n\to \infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}ρ_n ((k,k+1)),\] 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 ${\lambda }_{n,1}\ge {\lambda }_{n,2}\ge ...\ge 0$ is the partition of $n$ corresponding to ${\rho }_n$, ${\mu }_{n,1}\ge {\mu }_{n,2}\ge >...\ge 0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of ${\mu }_n$ is the transpose of the Young diagram of ${\lambda }_n$), ${\lim }_{n\to \infty }\frac{{\lambda }_{n,i}}{n}=p_i$ for each $i\ge 1$, and ${\lim }_{n\to \infty }\frac{{\mu }_{n,j}}{n}=q_j$ for each $j\ge 1$, 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 ${\sum }_i{p}_i^2-{\sum }_j{q}_j^2$ and $Z$ is a standard Gaussian random variable.

Published in at http://dx.doi.org/10.1214/08-AOP418 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Gaussian fluctuations of characters of symmetric groups and of Young diagrams