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

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 $0<γ_1<γ_2$ such that if $L(t)\ge e^{γt^{d/(d+2)}}$, with $γ>γ_1$, a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if $L(t)\ge e^{γt^{d/(d+2)}}$, with $γ>γ_2$, 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 $L(t)=e^{γt^{d/(d+2)}}$ with $γ<γ_2$: it is shown that they are infinitely divisible laws with a Lévy spectral function which explodes when $x\to0$ 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.

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