Attributing a positive value τx to each x∈ℤd, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (τx), often known as “Bouchaud’s trap model.” We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d≥5. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as the...

We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk...

We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition $\left(T^{\text{'}}\right)$. We show that for every $ϵ>0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $exp(-{(logn)}^{d-ϵ})$. This bound almost matches the known lower bound of $exp(-C{(logn)}^d)$, and significantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched probability...

Branching Processes in Random Environment (BPREs) $(Z_n:n\ge 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $\mathbb{P}(1\le Z_n\le k|Z_0=i)$, $k,i\in \mathbb{N}$ as $n\to \infty$. More precisely, we characterize the exponential...

We study trajectories of $d$-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential $V$ is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii $\nu$ has power-law decay and prove that superdiffusivity...

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) $\xi$ is strictly less than $1$ and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and...