Transition from the annealed to the quenched asymptotics for a random walk on random obstacles

Abstract

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 $L(t)$. T here are constants $0<{\gamma }_1<{\gamma }_2$ such that if $L(t)\ge e^{\gamma t^{d/(d+2)}}$, with $\gamma >{\gamma }_1$, a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if $L(t)\ge e^{\gamma t^{d/(d+2)}}$, with $\gamma >{\gamma }_2$, also a central limit theorem is satisfied. If $L(t)\ll t$, the averaged survival probability decreases like the quenched survival probability. If $t\ll L(t)$ and $\log L(t)\ll t^{d/(d+2)}$ we obtain an intermediate regime. Furthermore, when the dimension $d=1$ it is possible to describe the fluctuations of the averaged survival probability when $L(t)=e^{\gamma t^{d/(d+2)}}$ with $\gamma <{\gamma }_2$: it is shown that they are infinitely divisible laws with a Lévy spectral function which explodes when $x\to 0$ as stable laws of characteristic exponent $\alpha <2$. 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.

Published at http://dx.doi.org/10.1214/009117905000000404 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (13)

• Extreme value problems in Random Matrix Theory and other disordered systems
• Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials
• Sinai model in presence of dilute absorbers
• A Phase Transition in a Quenched Amorphous Ferromagnet
• Biased random walks on a Galton-Watson tree with leaves
• Large fluctuations and transport properties of the Lévy-Lorentz gas
• Complete localisation in the parabolic Anderson model with Pareto-distributed potential
• Limit theorems for time-dependent averages of nonlinear stochastic heat equations
• Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour
• Random Walk on Lattice with an Antisymmetric Perturbation in One Point
• Diffusion Approximations for $\text{Cox}/G_t/\infty$ Queues In A Fast Oscillatory Random Environment
• Survival time of random walk in random environment among soft obstacles
• Stable limit laws and structure of the scaling function for reaction-diffusion in random environment