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November 2005 Transition from the annealed to the quenched asymptotics for a random walk on random obstacles
Gérard Ben Arous, Stanislav Molchanov, Alejandro F. Ramírez
Ann. Probab. 33(6): 2149-2187 (November 2005). DOI: 10.1214/009117905000000404

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). There are constants 0<γ1<γ2 such that if L(t)≥eγtd/(d+2), with γ>γ1, a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if L(t)≥eγtd/(d+2), with γ>γ2, also a central limit theorem is satisfied. If L(t)≪t, the averaged survival probability decreases like the quenched survival probability. If tL(t) and logL(t)≪td/(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)=eγtd/(d+2) with γ<γ2: it is shown that they are infinitely divisible laws with a Lévy spectral function which explodes when x→0 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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Gérard Ben Arous. Stanislav Molchanov. Alejandro F. Ramírez. "Transition from the annealed to the quenched asymptotics for a random walk on random obstacles." Ann. Probab. 33 (6) 2149 - 2187, November 2005. https://doi.org/10.1214/009117905000000404

Information

Published: November 2005
First available in Project Euclid: 7 December 2005

zbMATH: 1099.82003
MathSciNet: MR2184094
Digital Object Identifier: 10.1214/009117905000000404

Subjects:
Primary: 82B41 , 82B44
Secondary: 60J45 , 60J65 , 82C22

Keywords: enlargement of obstacles , Parabolic Anderson model , Principal eigenvalue , Random walk , Wiener sausage

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 6 • November 2005
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