We prove that the edge-reinforced random walk on the ladder ℤ×{1,2} with initial weights a>3/4 is recurrent. The proof uses a known representation of the edge-reinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multicomponent Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyze this Gibbsian process. Furthermore, we prove spatially exponentially fast decreasing bounds for normalized local times of the edge-reinforced random walk on a finite piece of the ladder, uniformly in the size of the finite piece.

Qian, Luscombe and Gerstein [J. Molecular Biol. 313 (2001) 673–681] introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate 1. When a new individual is born, it has the same type as its parent with probability 1−r and is a new type, different from all previously observed types, with probability r. We refer to individuals with the same type as families and provide an approximation to the joint distribution of family sizes when the population size reaches N. We also show that if 1≪S≪N^1−r, then the number of families of size at least S is approximately CNS^−1/(1−r), while if N^1−r≪S the distribution decays more rapidly than any power.

The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side of the path from a point near the end of the current path. We prove that the harmonic explorer converges to SLE₄ as the grid gets finer.

In this work we study a natural transition mechanism describing the passage from a quenched (almost sure) regime to an annealed (in average) one, for a symmetric simple random walk on random obstacles on sites having an identical and independent law. The transition mechanism we study was first proposed in the context of sums of identical independent random exponents by Ben Arous, Bogachev and Molchanov in [Probab. Theory Related Fields 132 (2005) 579–612]. Let p(x,t) be the survival probability at time t of the random walk, starting from site x, and let L(t) be some increasing function of time. We show that the empirical average of p(x,t) over a box of side L(t) has different asymptotic behaviors depending on L(t). There are constants 0<γ₁<γ₂ such that if L(t)≥e^γtd/(d+2), with γ>γ₁, a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if L(t)≥e^γtd/(d+2), with γ>γ₂, also a central limit theorem is satisfied. If L(t)≪t, the averaged survival probability decreases like the quenched survival probability. If t≪L(t) and logL(t)≪t^d/(d+2) we obtain an intermediate regime. Furthermore, when the dimension d=1 it is possible to describe the fluctuations of the averaged survival probability when L(t)=e^γtd/(d+2) with γ<γ₂: it is shown that they are infinitely divisible laws with a Lévy spectral function which explodes when x→0 as stable laws of characteristic exponent α<2. These results show that the quenched and annealed survival probabilities correspond to a low- and high-temperature behavior of a mean-field type phase transition mechanism.

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton–Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams’ decomposition for Brownian motion with drift.

Let b be an integer greater than 1 and let W^ɛ=(W^ɛ_n;n≥0) be a random walk on the b-ary rooted tree $\mathbb {U}_{b}$, starting at the root, going up (resp. down) with probability 1/2+ɛ (resp. 1/2−ɛ), ɛ∈(0,1/2), and choosing direction i∈{1,…,b} when going up with probability a_i. Here a=(a₁,…,a_b) stands for some nondegenerated fixed set of weights. We consider the range {W^ɛ_n;n≥0} that is a subtree of $\mathbb {U}_{b}$. It corresponds to a unique random rooted ordered tree that we denote by τ_ɛ. We rescale the edges of τ_ɛ by a factor ɛ and we let ɛ go to 0: we prove that correlations due to frequent backtracking of the random walk only give rise to a deterministic phenomenon taken into account by a positive factor γ(a). More precisely, we prove that τ_ɛ converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by γ(a). We actually state the result in the more general case of a random walk on a tree with an infinite number of branches at each node (b=∞) and for a general set of weights a=(a_n,n≥0).

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on Z. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on Z^d and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on Z^d. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as t→∞ for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.

We show that if μ is an invariant measure for the long range exclusion process putting no mass on the full configuration, L is the formal generator of that process and f is a cylinder function, then Lf∈L¹(dμ) and ∫Lf dμ=0. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on ℤ^d when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on ℤ when the underlying random walk is irreducible and either has zero mean or allows jumps only to the nearest-neighbors.

We prove that the logarithmic Sobolev constant for zero-range processes in a box of diameter L grows as L².

We consider the following interacting particle system: There is a “gas” of particles, each of which performs a continuous-time simple random walk on ℤ^d, with jump rate D_A. These particles are called A-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with N_A(x,0−) A-particles at x, and that the N_A(x,0−),x∈ℤ^d, are i.i.d., mean-μ_A Poisson random variables. In addition, there are B-particles which perform continuous-time simple random walks with jump rate D_B. We start with a finite number of B-particles in the system at time 0. B-particles are interpreted as individuals who have heard a certain rumor or who are infected. The B-particles move independently of each other. The only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle.

We investigate how fast the rumor, or infection, spreads. Specifically, if B̃(t):={x∈ℤ^d: a B-particle visits x during [0,t]} and B(t)=B̃(t)+[−1/2,1/2]^d, then we investigate the asymptotic behavior of B(t). Our principal result states that if D_A=D_B (so that the A- and B-particles perform the same random walk), then there exist constants 0<C_i<∞ such that almost surely $\mathcal{C}(C_{2}t)\subset B(t)\subset \mathcal{C}(C_{1}t)$ for all large t, where $\mathcal{C}(r)=[-r,r]^{d}$. In a further paper we shall use the results presented here to prove a full “shape theorem,” saying that t⁻¹B(t) converges almost surely to a nonrandom set B₀, with the origin as an interior point, so that the true growth rate for B(t) is linear in t.

If D_A≠D_B, then we can only prove the upper bound $B(t)\subset \mathcal{C}(C_{1}t)$ eventually.