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We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region $D$ is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.Comment: Published at http://dx.doi.org/10.1214/154957806000000078 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Given an n × n complex matrix A, letbe the empirical spectral distribution (ESD) of its eigenvalues λ i ∈ C, i = 1, . . . n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD µ 1 √ n An of a random matrix A n = (a ij ) 1≤i,j≤n where the random variables a ij − E(a ij ) are iid copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real of complex gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1

Abstract. Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

We study the empirical measure LA n of the eigenvalues of nonnormal square matrices of the form An = UnTnVn with Un, Vn independent Haar distributed on the unitary group and Tn real diagonal. We show that when the empirical measure of the eigenvalues of Tn converges, and Tn satisfies some technical conditions, LA n converges towards a rotationally invariant measure µ on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem [6]. We also consider the case where Un, Vn are independently Haar distributed on the orthogonal group.