Mathematics – Probability
Scientific paper
2007-08-13
Annals of Probability 2009, Vol. 37, No. 2, 726-741
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/08-AOP418 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
10.1214/08-AOP418
We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_n ((k,k+1)),\] 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 $\lambda_{n,1}\ge\lambda_{n,2}\ge...\ge0$ is the partition of $n$ corresponding to $\rho_n$, $\mu_{n,1}\ge\mu_{n,2}\ge >...\ge0$ 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 $i\ge1$, and $\lim_{n\to\infty}\frac{\mu_{n,j}}{n}=q_j$ for each $j\ge1$, 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 $\sum_ip_i^2-\sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.
No associations
LandOfFree
Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-388458