The one-dimensional nearest-neighbor totally asymmetric simple exclusion process can be constructed in the same space as a last-passage percolation model in ℤ². We show that the trajectory of a second class particle in the exclusion process can be linearly mapped into the competition interface between two growing clusters in the last-passage percolation model. Using technology built up for geodesics in percolation, we show that the competition interface converges almost surely to an asymptotic random direction. As a consequence we get a new proof for the strong law of large numbers for the second class particle in the rarefaction fan and describe the distribution of the asymptotic angle of the competition interface.

Let ℓ be the projected intersection local time of two independent Brownian paths in ℝ^d for d=2,3. We determine the lower tail of the random variable $\ell(\mathbb {U})$, where $\mathbb {U}$ is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.

Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T)=0 conditioned to stay above the semicircle $c_{T}(t)=\sqrt{T^{2}-t^{2}}$. In the limit of large T, the fluctuation scale of b(t)−c_T(t) is T^1/3 and its time-correlation scale is T^2/3. We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t=τT, τ∈(−1,1), is only through the second derivative of c_T(t) at t=τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T^γ, γ>1/2. The fluctuation scale is then T^(2−γ)/3. More general conditioning shapes are briefly discussed.

In a celebrated paper, Dyson shows that the spectrum of an n×n random Hermitian matrix, diffusing according to an Ornstein–Uhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled.

In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy process at two different times. Similarly we find a PDE satisfied by the joint distribution of the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process, which itself is based on the joint probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE for the Dyson process is then subjected to an asymptotic analysis, consistent with the edge and bulk rescalings. The PDEs enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t₁ and t₂, when t₂−t₁→∞, as illustrated in this paper for the Airy process. This paper also contains a rigorous proof that the extended Hermite kernel, governing the joint probabilities for the Dyson process, converges to the extended Airy and Sine kernels after the appropriate rescalings.

We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process f. Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation. We also give a lower bound on this exponential rate of decay in terms of the maximal variance of a family of Gaussian processes f^x, derived from the original process f.

We investigate the Gaussian small ball probabilities with random centers, find their deterministic a.s.-equivalents and establish a relation to infinite-dimensional high-resolution quantization.

We consider the empirical process $\mathbb{G}_{t}$ of a one-dimensional diffusion with finite speed measure, indexed by a collection of functions ℱ. By the central limit theorem for diffusions, the finite-dimensional distributions of $\mathbb{G}_{t}$ converge weakly to those of a zero-mean Gaussian random process $\mathbb{G}$. We prove that the weak convergence $\mathbb{G}_{t}\Rightarrow \mathbb{G}$ takes place in ℓ^∞(ℱ) if and only if the limit $\mathbb{G}$ exists as a tight, Borel measurable map. The proof relies on majorizing measure techniques for continuous martingales. Applications include the weak convergence of the local time density estimator and the empirical distribution function on the full state space.

Motivated by the recent work of Benjamini, Häggström, Peres and Steif [Ann. Probab. 34 (2003) 1–34] on dynamical random walks, we do the following: (i) Prove that, after a suitable normalization, the dynamical Gaussian walk converges weakly to the Ornstein–Uhlenbeck process in classical Wiener space; (ii) derive sharp tail-asymptotics for the probabilities of large deviations of the said dynamical walk; and (iii) characterize (by way of an integral test) the minimal envelope(s) for the growth-rate of the dynamical Gaussian walk. This development also implies the tail capacity-estimates of Mountford for large deviations in classical Wiener space.

The results of this paper give a partial affirmative answer to the problem, raised in Benjamini, Häggström, Peres and Steif [Ann. Probab. 34 (2003) 1–34, Question 4], of whether there are precise connections between the OU process in classical Wiener space and dynamical random walks.

This paper is devoted to the problem of sample path large deviations for the Markov processes on ℝ₊^N having a constant but different transition mechanism on each boundary set {x:x_i=0 for i∉Λ, x_i>0 for i∈Λ}. The global sample path large deviation principle and an integral representation of the rate function are derived from local large deviation estimates. Our results complete the proof of Dupuis and Ellis of the sample path large deviation principle for Markov processes describing a general class of queueing networks.

We investigate the evolution of barycenters of masses transported by stochastic flows. The state spaces under consideration are smooth affine manifolds with certain convexity structure. Under suitable conditions on the flow and on the initial measure, the barycenter {Z_t} is shown to be a semimartingale and is described by a stochastic differential equation. For the hyperbolic space the barycenter of two independent Brownian particles is a martingale and its conditional law converges to that of a Brownian motion on the limiting geodesic. On the other hand for a large family of discrete measures on suitable Cartan–Hadamard manifolds, the barycenter of the measure carried by an unstable Brownian flow converges to the Busemann barycenter of the limiting measure.

We give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the Chen–Souriau sense and to establish a link with cyclic cohomology. With the second one, we can establish a stochastic fixed point theorem.

We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase transition takes various forms, depending on the values of the parameters controlling the different types of hyperedges. It may be continuous as in a random graph. (In fact, when there are no higher-order edges, it is exactly the emergence of the giant component.) In this case, there is a sequence of possible sizes of “components” (including but not restricted to N^2/3). Alternatively, the phase transition may be discontinuous. We are particularly interested in the nature of the discontinuous phase transition and are able to exhibit precise asymptotics. Our method extends a result of Aldous [Ann. Probab. 25 (1997) 812–854] on component sizes in a random graph.

Let {X,X_n;n≥1} be a sequence of i.i.d. mean-zero random variables, and let S_n=∑_i=1ⁿX_i,n≥1. We establish necessary and sufficient conditions for having with probability 1, 0<lim sup _n→∞|S_n|/$\sqrt{nh(n)}$<∞, where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n)=(loglogn)^p, where p>1 and to h(n)=(logn)^r, r>0, we obtain analogues of the Hartman–Wintner LIL in the infinite variance case. Our proof is based on a general result dealing with LIL behavior of the normalized sums {S_n/c_n;n≥1}, where c_n is a sufficiently regular normalizing sequence.

In this paper, we are interested in the behavior of the typical Poisson–Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity. We prove a law of large numbers for its number of vertices and the area of the cell outside the disk. Moreover, for the latter, we establish a central limit theorem as well as moderate deviation type results. The proofs deeply rely on precise connections between Poisson–Voronoi tessellations, convex hulls of Poisson samples and germ–grain models in the unit ball. Besides, we derive analogous facts for the Crofton cell of a stationary Poisson line process in the plane.