In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices (M_N)_N defined by $M_{N}=W_{N}/\sqrt{N}+A_{N}$ where W_N is an N×N Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix A_N is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of A_N are sufficiently far from zero, the corresponding eigenvalues of M_N almost surely exit the limiting semicircle compact support as the size N becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of W_N. On the other hand, when A_N is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of W_N.

We consider an interacting particle system (η_t)_t≥0 with values in {0, 1}^ℤ, in which each vacant site becomes occupied with rate 1, while each connected component of occupied sites become vacant with rate equal to its size. We show that such a process admits a unique invariant distribution, which is exponentially mixing and can be perfectly simulated. We also prove that for any initial condition, the avalanche process tends to equilibrium exponentially fast, as time increases to infinity. Finally, we consider a related mean-field coagulation–fragmentation model, we compute its invariant distribution and we show numerically that it is very close to that of the interacting particle system.

In this paper, we look at the properties of limits of a sequence of real valued inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions, then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as the almost-continuous diffusions. These processes are strong Markov and satisfy an “almost-continuity” condition. We also give a simple condition for the limit to be a continuous diffusion.

These results contrast with the multidimensional case where, as we show with an example, a sequence of two-dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov.

In this paper, we study the bead model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire; there must be exactly one bead on each neighboring wire. We construct a one-parameter family of Gibbs measures on the bead configurations that are uniform in a certain sense. When endowed with one of these measures, this model is shown to be a determinantal point process, whose marginal on each wire is the sine process (given by eigenvalues of large hermitian random matrices). We prove then that this process appears as a limit of any dimer model on a planar bipartite graph when some weights degenerate.

We consider a nearest-neighbor, one dimensional random walk {X_n}_n≥0 in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that X_n is of order n^s for some s<1. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences {n_k} and {x_k} depending on the environment only, such that X_nk−x_k=o(log n_k)² (a localized regime). On the other hand, there exist sequences {t_m} and {s_m} depending on the environment only, such that logs_m/log t_m→s<1 and P_ω(X_tm/s_m≤x)→1/2 for all x>0 and →0 for x≤0 (a spread out regime).

We consider random walk (X_n)_n≥0 on ℤ^d in a space–time product environment ω∈Ω. We take the point of view of the particle and focus on the environment Markov chain (T_n, Xnω)_n≥0 where T denotes the shift on Ω. Conditioned on the particle having asymptotic mean velocity equal to any given ξ, we show that the empirical process of the environment Markov chain converges to a stationary process μ_ξ^∞ under the averaged measure. When d≥3 and ξ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity ξ, the empirical process of the environment Markov chain converges to μ_ξ^∞ under the quenched measure as well. In this case, we show that μ_ξ^∞ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob h-transform.

In this paper the solutions u_ν=u_ν(x, t) to fractional diffusion equations of order 0<ν≤2 are analyzed and interpreted as densities of the composition of various types of stochastic processes.

For the fractional equations of order ν=1/2ⁿ, n≥1, we show that the solutions u_1/2ⁿ correspond to the distribution of the n-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order ν=2/3ⁿ, n≥1, is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions.

In the general case we show that u_ν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u_ν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

Properties of the law μ of the integral ∫₀^∞c^−N_t− dY_t are studied, where c>1 and {(N_t, Y_t), t≥0} is a bivariate Lévy process such that {N_t} and {Y_t} are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein–Uhlenbeck process. The law μ is parametrized by c, q and r, where p=1−q−r, q, and r are the normalized Lévy measure of {(N_t, Y_t)} at the points (1, 0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p>0 and q>0, μ_c, q, r is infinitely divisible if and only if r≤pq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r=1. It is shown that if c is in the set of Pisot–Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q>0. On the other hand, for Lebesgue almost every c>1, there are positive constants C₁ and C₂ such that μ is absolutely continuous whenever q≥C₁p≥C₂r. For any c>1 there is a positive constant C₃ such that μ is continuous-singular whenever q>0 and max {q, r}≤C₃p. Here, if {N_t} and {Y_t} are independent, then r=0 and q=b/(a+b).

The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under nondegeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure.

We consider two families of random matrix-valued analytic functions: (1) G₁−zG₂ and (2) G₀+zG₁+z²G₂+⋯, where G_i are n×n random matrices with independent standard complex Gaussian entries. The random set of z where these matrix-analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain Hilbert spaces (“Bargmann–Fock spaces”) of holomorphic functions on the corresponding surfaces. Along with the new results, this also gives a unified framework in which to view a theorem of Peres and Virág (n=1 in the second setting above) and a well-known result of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).

The parabolic Anderson problem is the Cauchy problem for the heat equation ∂_tu(t, z)=Δu(t, z)+ξ(z)u(t, z) on (0, ∞)×ℤ^d with random potential (ξ(z):z∈ℤ^d). We consider independent and identically distributed potentials, such that the distribution function of ξ(z) converges polynomially at infinity. If u is initially localized in the origin, that is, if $u(0,{z})=𝟙_{0}({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.

It is shown that the ratio between the expected diameter of an L₂-bounded martingale and the standard deviation of its last term cannot exceed $\sqrt{3}$. Moreover, a one-parameter family of stopping times on standard Brownian motion is exhibited, for which the $\sqrt{3}$ upper bound is attained. These stopping times, one for each cost-rate c, are optimal when the payoff for stopping at time t is the diameter D(t) obtained up to time t minus the hitherto accumulated cost ct. A quantity related to diameter, maximal drawdown (or rise), is introduced and its expectation is shown to be bounded by $\sqrt{2}$ times the standard deviation of the last term of the martingale. These results complement the Dubins and Schwarz respective bounds 1 and $\sqrt{2}$ for the ratios between the expected maximum and maximal absolute value of the martingale and the standard deviation of its last term. Dynamic programming (gambling theory) methods are used for the proof of optimality.