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

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.Comment: 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

Subtree prune and regraft: a reversible real tree-valued Markov process

We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov--Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.Comment: Published at http://dx.doi.org/10.1214/009117906000000034 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org