The motivation for this paper is the study of the phase transition for recurrence/ transience of a class of self‐interacting random walks on trees, which includes the once‐reinforced random walk. For this purpose, we define a quantity, which we call the branching‐ruin number of a tree, which provides (in the spirit of Furstenberg [11] and Lyons [13]) a natural way to measure trees with polynomial growth. We prove that the branching‐ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once‐reinforced random walk. We define a sharp and effective (i.e., computable) criterion characterizing the recurrence/transience of a larger class of self‐interacting walks on trees, providing the complete picture for their phase transition. © 2019 Wiley Periodicals, Inc.

We consider elliptic random walks in i.i.d. random environments on Z d . The main goal of this paper is to study under which ellipticity conditions local trapping occurs. Our main result is to exhibit an ellipticity criterion for ballistic behavior which extends previously known results. We also show that if the annealed expected exit time of a unit hypercube is infinite then the walk has zero asymptotic velocity.

No abstract

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $$\rho \in [0, 1]$$ ρ ∈ [ 0 , 1 ] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $$\rho $$ ρ except for at most two values $$\rho _-, \rho _+ \in [0, 1]$$ ρ - , ρ + ∈ [ 0 , 1 ] . The asymptotic speed we obtain in our LLN is a monotone function of $$\rho $$ ρ . Also, $$\rho _-$$ ρ - and $$\rho _+$$ ρ + are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

In this short paper, we consider the Once-reinforced random walk with reinforcement parameter a on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter a0 such that the Once-reinforced random walk is almost surely recurrent if a > a0 and almost surely transient if a < a0. This provides the first examples of phase transition for the Once-reinforced random walk.