We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large N the disconnection time of G_N×ℤ has rough order |G_N|², when G_N=(ℤ/Nℤ)^d. In agreement with a conjecture by I. Benjamini, we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree.

Based on an analytical approach to the definition of additive free convolution on probability measures on the real line, we prove free analogues of limit theorems for sums for nonidentically distributed random variables in classical probability theory.

We study two variants of the notion of holes formed by planar simple random walk of time duration 2n and the areas associated with them. We prove in both cases that the number of holes of area greater than A(n), where {A(n)} is an increasing sequence, is, up to a logarithmic correction term, asymptotic to n⋅A(n)⁻¹ for a range of large holes, thus confirming an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to n. We also discuss the different exponent of 5/3 observed by Mandelbrot for small holes.

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes …, X₋₁, X₀, X₁, … whose partial sums S_n=X₁+⋯+X_n are of the form S_n=M_n+R_n, where M_n is a square integrable martingale with stationary increments and R_n is a remainder term for which E(R_n²)=o(n). Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting ‖⋅‖ denote the norm in L²(P), a sufficient condition for the partial sums of a stationary process to have the form S_n=M_n+R_n is that n^−3/2‖E(S_n|X₀, X₋₁, …)‖ be summable. A sufficient condition for the LIL is only slightly stronger, requiring n^−3/2log^3/2(n)‖E(S_n|X₀, X₋₁, …)‖ to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.

It is shown that functions defined on {0, 1, …, r−1}ⁿ satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger “local” sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand’s [Ann. Probab. 22 (1994) 1576–1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on {0, 1, …, r−1}ⁿ for r≥2.

We wish to characterize when a Lévy process X_t crosses boundaries like t^κ, κ>0, in a one- or two-sided sense, for small times t; thus, we inquire when lim sup _t↓0|X_t|/t^κ, lim sup _t↓0X_t/t^κ and/or lim inf _t↓0X_t/t^κ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of κ>0. This completes and extends a line of research going back to Blumenthal and Getoor in the 1960s. Often (for many values of κ), when the lim sups are finite a.s., they are in fact zero, but the lim sups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one- or two-sided boundaries in quite different ways, but surprisingly this is not so for the case κ=1/2, where a new kind of analogue of an iterated logarithm law with a square root boundary is derived. An integral test is given to distinguish the possibilities in that case.

Sinai’s walk can be thought of as a random walk on ℤ with random potential V, with V weakly converging under diffusive rescaling to a two-sided Brownian motion. We consider here the generator $\mathbb {L}_{N}$ of Sinai’s walk on [−N, N]∩ℤ with Dirichlet conditions on −N, N. By means of potential theory, for each h>0, we show the relation between the spectral properties of $\mathbb {L}_{N}$ for eigenvalues of order $o(\exp(-h\sqrt{N}))$ and the distribution of the h-extrema of the rescaled potential $V_{N}(x)\equiv V(Nx)/\sqrt{N}$ defined on [−1, 1]. Information about the h-extrema of V_N is derived from a result of Neveu and Pitman concerning the statistics of h-extrema of Brownian motion. As first application of our results, we give a proof of a refined version of Sinai’s localization theorem.

This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated Markov jump processes are analyzed under a thermodynamic limit regime, that is, when the networks have some symmetry properties and when the number of nodes goes to infinity. An intriguing stability property is proved: under some conditions on the parameters, it is shown that, in the limit, several stable equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Networks with a unique equilibrium point are also presented.

We consider a diffusion process X in a random Lévy potential $\mathbb{V}$ which is a solution of the informal stochastic differential equation $$\begin{eqnarray*}\cases{dX_{t}=d\beta_{t}-\frac{1}{2}\mathbb{V}'(X_{t})\,dt,\cr X_{0}=0,}\end{eqnarray*}$$ (β B. M. independent of $\mathbb{V}$). We study the rate of convergence when the diffusion is transient under the assumption that the Lévy process $\mathbb{V}$ does not possess positive jumps. We generalize the previous results of Hu–Shi–Yor for drifted Brownian potentials. In particular, we prove a conjecture of Carmona: provided that there exists 0<κ<1 such that $\mathbf{E}[e^{\kappa\mathbb{V}_{1}}]=1$, then X_t/t^κ converges to some nondegenerate distribution. These results are in a way analogous to those obtained by Kesten–Kozlov–Spitzer for the transient random walk in a random environment.

We establish a general formula for the Laplace transform of the hitting times of a Gaussian process. Some consequences are derived, and particular cases like the fractional Brownian motion are discussed.

We consider the first passage percolation model on Z². In this model, we assign independently to each edge e a passage time t(e) with a common distribution F. Let T(u, v) be the passage time from u to v. In this paper, we show that, whenever F(0)<p_c, σ²(T((0, 0), (n, 0)))≥C log n for all n≥1. Note that if F satisfies an additional special condition, inf supp (F)=r>0 and F(r)>p⃗_c, it is known that there exists M such that for all n, σ²(T((0, 0), (n, n)))≤M. These results tell us that shape fluctuations not only depend on distribution F, but also on direction. When showing this result, we find the following interesting geometrical property. With the special distribution above, any long piece with r-edges in an optimal path from (0, 0) to (n, 0) has to be very circuitous.

We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.