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Transition from the annealed to the quenched asymptotics for a random walk on random obstacles | | Gerard Ben Arous
; Stanislav Molchanov
; Alejandro F. Ramirez
; | | Date: |
7 Dec 2004 | | Subject: | Probability MSC-class: 82B41;82B44 | math.PR | | 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 cite{bbm}. Let $p(x,t)$ be the survival probability at time $t$ of the random walk, starting from site $x$, and $L(t)$ 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 $0gamma_1$, 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 $tll L(t)$ 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 | | Source: | arXiv, math.PR/0501107 | | Services: | Forum | Review | PDF | Favorites | |
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