The Annals of Probability

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

Gérard Ben Arous, Stanislav Molchanov, and Alejandro F. Ramírez

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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.

Article information

Ann. Probab. Volume 33, Number 6 (2005), 2149-2187.

First available in Project Euclid: 7 December 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J65: Brownian motion [See also 58J65] 82C22: Interacting particle systems [See also 60K35]

Parabolic Anderson model random walk enlargement of obstacles principal eigenvalue Wiener sausage


Ben Arous, Gérard; Molchanov, Stanislav; Ramírez, Alejandro F. Transition from the annealed to the quenched asymptotics for a random walk on random obstacles. Ann. Probab. 33 (2005), no. 6, 2149--2187. doi:10.1214/009117905000000404.

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