We consider a discrete elliptic equation on the d-dimensional lattice ℤ^d with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric “homogenized” matrix A_hom=a_hom Id is characterized by ξ⋅A_homξ=〈(ξ+∇ϕ)⋅A(ξ+∇ϕ)〉 for any direction ξ∈ℝ^d, where the random field ϕ (the “corrector”) is the unique solution of −∇^∗⋅A(ξ+∇ϕ)=0 such that ϕ(0)=0, ∇ϕ is stationary and 〈∇ϕ〉=0, 〈⋅〉 denoting the ensemble average (or expectation).

It is known (“by ergodicity”) that the above ensemble average of the energy density $\mathcal {E}=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of $\mathcal {E}$ on length scales L satisfies the optimal estimate, that is, $\operatorname{var}[\sum \mathcal {E}\eta_{L}]\lesssim L^{-d}$, where the averaging function [i.e., ∑η_L=1, supp(η_L)⊂{|x|≤L}] has to be smooth in the sense that |∇η_L|≲L^−1−d. In two space dimensions (i.e., d=2), there is a logarithmic correction. This estimate is optimal since it shows that smooth averages of the energy density $\mathcal {E}$ decay in L as if $\mathcal {E}$ would be independent from edge to edge (which it is not for d>1).

This result is of practical significance, since it allows to estimate the dominant error when numerically computing a_hom.

We show that Talagrand’s transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand’s inequality is stable under bounded perturbations.

We study the probability distribution of the area and the number of vertices of random polygons in a convex set K⊂ℝ². The novel aspect of our approach is that it yields uniform estimates for all convex sets K⊂ℝ² without imposing any regularity conditions on the boundary ∂K. Our main result is a central limit theorem for both the area and the number of vertices, settling a well-known conjecture in the field. We also obtain asymptotic results relating the growth of the expectation and variance of these two functionals.

In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for noncommutative random variables. These notions are related to the liberation process introduced by Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.

Representations of branching Markov processes and their measure-valued limits in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. Each particle has a location and a “level,” but unlike earlier constructions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level r, or for the limiting models, hits infinity. For branching Markov processes, at each time t, conditioned on the state of the process, the levels are independent and uniformly distributed on [0, r]. For the limiting measure-valued process, at each time t, the joint distribution of locations and levels is conditionally Poisson distributed with mean measure K(t)×Λ, where Λ denotes Lebesgue measure, and K is the desired measure-valued process.

The representation simplifies or gives alternative proofs for a variety of calculations and results including conditioning on extinction or nonextinction, Harris’s convergence theorem for supercritical branching processes, and diffusion approximations for processes in random environments.

We show that for a typical coordinate projection of a subgaussian class of functions, the infimum over signs inf_(εi) sup_f∈F|∑_i=1^k ε_if(X_i)| is asymptotically smaller than the expectation over signs as a function of the dimension k, if the canonical Gaussian process indexed by F is continuous. To that end, we establish a bound on the discrepancy of an arbitrary subset of ℝ^k using properties of the canonical Gaussian process the set indexes, and then obtain quantitative structural information on a typical coordinate projection of a subgaussian class.

We study the Wiener–Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener–Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as infinite series representations and asymptotic expansions for the density of supremum, explicit expressions for the Wiener–Hopf factors and the Mellin transform of the supremum, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional.

This note is devoted to construct a rough path above a multidimensional fractional Brownian motion B with any Hurst parameter H∈(0, 1), by means of its representation as a Volterra Gaussian process. This approach yields some algebraic and computational simplifications with respect to [Stochastic Process. Appl. 120 (2010) 1444–1472], where the construction of a rough path over B was first introduced.

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs (G_n)_n≥0 converging locally to a Galton–Watson tree T (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function φ_∗ of T. In the first part, we show that the adjacency operator associated with T is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on φ_∗ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of (G_n)_n≥0. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.

We construct a bounded degree graph G, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also prove that the expected number of cutpoints of any transient Markov chain is infinite. This answers two questions of James, Lyons and Peres [A Transient Markov Chain With Finitely Many Cutpoints (2007) Festschrift for David Freedman].

Additionally, we consider a simple random walk on a finite connected graph G that starts at some fixed vertex x and is stopped when it first visits some other fixed vertex y. We provide a lower bound on the expected effective resistance between x and y in the path of the walk, giving a partial answer to a question raised in [Ann. Probab. 35 (2007) 732–738].

We study the problem of finding an universal estimation scheme h_n : ℝⁿ→ℝ, n=1, 2, … which will satisfy

lim_t→∞(1/t)∑_i=1^t|h_i(X₀, X₁, …, X_i−1)−E(X_i|X₀, X₁, …, X_i−1)|^p=0 a.s.

for all real valued stationary and ergodic processes that are in L^p. We will construct a single such scheme for all 1<p≤∞, and show that for p=1 mere integrability does not suffice but L log⁺ L does.

We study time-inhomogeneous Markov chains with finite state spaces using Nash and logarithmic-Sobolev inequalities, and the notion of c-stability. We develop the basic theory of such functional inequalities in the time-inhomogeneous context and provide illustrating examples.