Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

In this paper we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension $n$. In these cases the limit measure is given by the Marchenko-Pastur law and the semicircle law, respectively. For the weighted spectral measure we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.Comment: 23 page

A functional CLT for partial traces of random matrices

In this paper we show a functional central limit theorem for the sum of the first $\lfloor t n \rfloor$ diagonal elements of $f(Z)$ as a function in $t$, for $Z$ a random real symmetric or complex Hermitian $n\times n$ matrix. The result holds for orthogonal or unitarily invariant distributions of $Z$, in the cases when the linear eigenvalue statistic $\operatorname{tr} f(Z)$ satisfies a CLT. The limit process interpolates between the fluctuations of individual matrix elements as $f(Z)_{1,1}$ and of the linear eigenvalue statistic. It can also be seen as a functional CLT for processes of randomly weighted measures

Matrix measures, random moments and Gaussian ensembles

We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$ real or complex matrix measures defined on the interval $[0,1]$. The asymptotic properties of the first $k$ components of a uniformly distributed vector $(S_{1,n}, ..., S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n \to \infty$. In particular, it is shown that an appropriately centered and standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$ converges weakly to a vector of $k$ independent $p \times p$ Gaussian ensembles. For the proof of our results we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first $k$ canonical moments corresponding to the uniform distribution on the real or complex moment space $\mathcal{M}_n$ are independent multivariate Beta distributed random variables and that each of these random variables converge in distribution (if the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.Comment: 25 page

Distributions on unbounded moment spaces and random moment sequences

In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marchenko-Pastur distribution and Wigner's semi-circle law.Comment: Published in at http://dx.doi.org/10.1214/11-AOP693 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The speed of biased random walk among random conductances

We consider biased random walk among iid, uniformly elliptic conductances on $\mathbb{Z}^d$, and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 3: it follows along the lines of the proof of the Einstein relation in [GGN]. On the other hand, we give a counterexample showing that for iid, uniformly elliptic conductances, the velocity is not always increasing as a function of the bias. More precisely, if $d=2$ and if the conductances take the values $1$ (with probability $p$) and $\kappa$ (with probability $1-p$) and $p$ is close enough to $1$ and $\kappa$ small enough, the velocity is not increasing as a function of the bias, see Theorem 2

Sum rules and large deviations for spectral matrix measures

A sum rule relative to a reference measure on R is a relationship between the reversed Kullback-Leibler divergence of a positive measure on R and some non-linear functional built on spectral elements related to this measure (see for example Killip and Simon 2003). In this paper, using only probabilistic tools of large deviations, we extend the sum rules obtained in Gamboa, Nagel and Rouault (2015) to the case of Hermitian matrix-valued measures. We recover the earlier result of Damanik, Killip and Simon (2010) when the reference measure is the (matrix-valued) semicircle law and obtain a new sum rule when the reference measure is the (matrix-valued) Marchenko-Pastur law