We generalize Lindeberg’s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.

It has been recently shown that if X is an n×N matrix whose entries are i.i.d. standard complex Gaussian and l₁ is the largest eigenvalue of X^*X, there exist sequences m_n,N and s_n,N such that (l₁−m_n,N)/s_n,N converges in distribution to W₂, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W₂ is denoted F₂.

In this paper we show that, under the assumption that n/N→ γ∈(0, ∞), we can find a function M, continuous and nonincreasing, and sequences μ̃_n,N and σ̃_n,N such that, for all real s₀, there exists an integer N(s₀, γ) for which, if (n∧N)≥N(s₀, γ), we have, with l_n,N=(l₁−μ̃_n,N)/σ̃_n,N,

∀ s≥s₀ (n∧N)^2/3|P(l_n,N≤s)−F₂(s)|≤M(s₀)exp(−s).

The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W₂ is a good approximation to the empirical distribution of l_n,N in simulations, an important fact from the point of view of (e.g., statistical) applications.

Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting N→∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type X_N(f)=∑_k=1^Nf(λ_k) where λ₁, λ₂, …, λ_N denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533–605], where the analogous result for random sample covariance matrices is established.

Consider q_n a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper we show that, when n goes to +∞, q_n suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of the Brownian map is defined. The weak convergences hold in these metric spaces.

A class of random discrete distributions P is introduced by means of a recursive splitting of unity. Assuming supercritical branching, we show that for partitions induced by sampling from such P a power growth of the number of blocks is typical. Some known and some new partition structures appear when P is induced by a Dirichlet splitting.

In this paper, we study intermittency for the parabolic Anderson equation ∂u/∂t=κΔu+ξu, where u:ℤ^d×[0, ∞)→ℝ, κ is the diffusion constant, Δ is the discrete Laplacian and ξ:ℤ^d×[0, ∞)→ℝ is a space-time random medium. We focus on the case where ξ is γ times the random medium that is obtained by running independent simple random walks with diffusion constant ρ starting from a Poisson random field with intensity ν. Throughout the paper, we assume that κ, γ, ρ, ν∈(0, ∞). The solution of the equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ.

We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of u, and show that they display an interesting dependence on the dimension d and on the parameters κ, γ, ρ, ν, with qualitatively different intermittency behavior in d=1, 2, in d=3 and in d≥4. Special attention is given to the asymptotics of these Lyapunov exponents for κ↓0 and κ→∞.

We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes which, in the real-valued case, can be replaced by simpler norms of the kernel matrix (i.e., norms of some multilinear operators associated with the kernel matrix). As a corollary, we derive tail inequalities for U-statistics with bounded kernels and for some multiple stochastic integrals.

We derive two-sided estimates on moments and tails of Gaussian chaoses, that is, random variables of the form ∑a_i1,…,idg_i1⋯g_id, where g_i are i.i.d. ${\mathcal{N}}(0,1)$ r.v.’s. Estimates are exact up to constants depending on d only.

We consider Bernoulli percolation on a locally finite quasi-transitive unimodular graph and prove that two infinite clusters cannot have infinitely many pairs of vertices at distance 1 from one another or, in other words, that such graphs exhibit “cluster repulsion.” This partially answers a question of Häggström, Peres and Schonmann.

We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with p_c=1 on some nonunimodular graph. Such trees cannot exist for nonamenable unimodular graphs. We show a new way of constructing nonunimodular graphs that have properties more peculiar than the ones previously known.

We show that for the mean zero simple exclusion process in ℤ^d and for the asymmetric simple exclusion process in ℤ^d for d≥3, the self-diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices. The proof depends on a similar stability property of the Sobolev inner product associated with the operator.

We consider two species of particles performing random walks in a domain in ℝ^d with reflecting boundary conditions, which annihilate on contact. In addition, there is a conservation law so that the total number of particles of each type is preserved: When the two particles of different species annihilate each other, particles of each species, chosen at random, give birth. We assume initially equal numbers of each species and show that the system has a diffusive scaling limit in which the densities of the two species are well approximated by the positive and negative parts of the solution of the heat equation normalized to have constant L¹ norm. In particular, the higher Neumann eigenfunctions appear as asymptotically stable states at the diffusive time scale.

We give lower bounds for the density p_T(x, y) of the law of X_t, the solution of dX_t=σ(X_t) dB_t+b(X_t) dt, X₀=x, under the following local ellipticity hypothesis: there exists a deterministic differentiable curve x_t, 0≤t≤T, such that x₀=x, x_T=y and σσ^*(x_t)>0, for all t∈[0, T]. The lower bound is expressed in terms of a distance related to the skeleton of the diffusion process. This distance appears when we optimize over all the curves which verify the above ellipticity assumption.

The arguments which lead to the above result work in a general context which includes a large class of Wiener functionals, for example, Itô processes. Our starting point is work of Kohatsu-Higa which presents a general framework including stochastic PDE’s.