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

*(English)*Zbl 1099.82003In the paper a natural transition mechanism describing the passage from a quenched regime to an annealed one, for a symmetric random walk on random obstacles on sites having an identical and independent law, is studied. An argument of the transition mechanism used in the paper was firstly proposed in [B. Arous, L. Bogachev and S. Molchanov, Probab. Theory Relat. Fields 132, 579–612 (2005; Zbl 1073.60017)].

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. In the paper it is studied the averaged quantity \( p^{L(t)}(0,t)=\frac{1}{| \Lambda_{L(t)}| }\sum_{x\in\Lambda_{L(t)}}p(x,t), \) where \(\Lambda_{L(t)}=[-(2L(t)+1,(2L(t)+1)]^d\cap \mathbb{Z}^d\).

It is shown that \(p^{L(t)}(0,t)\) has different asymptotic behaviors depending on \(L(t)\). Namely, there are constants \(0<\gamma_1<\gamma_2\) such that if \(L(t)\geq e^{\gamma t^{d/(d+2)}}\), with \(\gamma>\gamma_1\), a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if \(L(t)\geq e^{\gamma t^{d/(d+2)}}\), with \(\gamma>\gamma_2\), also a central limit theorem is satisfied. If \(L(t)\ll t\), \(p^{L(t)}(0,t)\) decreases like the quenched survival probability. If \(t\ll L(t)\) and \(\log L(t)\ll t^{d/(d+2)}\) an intermediate regime is obtained.

Furthermore, when the dimension \(d=1\) it is possible to describe the fluctuations of \(p^{L(t)}(0,t)\) 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.

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. In the paper it is studied the averaged quantity \( p^{L(t)}(0,t)=\frac{1}{| \Lambda_{L(t)}| }\sum_{x\in\Lambda_{L(t)}}p(x,t), \) where \(\Lambda_{L(t)}=[-(2L(t)+1,(2L(t)+1)]^d\cap \mathbb{Z}^d\).

It is shown that \(p^{L(t)}(0,t)\) has different asymptotic behaviors depending on \(L(t)\). Namely, there are constants \(0<\gamma_1<\gamma_2\) such that if \(L(t)\geq e^{\gamma t^{d/(d+2)}}\), with \(\gamma>\gamma_1\), a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if \(L(t)\geq e^{\gamma t^{d/(d+2)}}\), with \(\gamma>\gamma_2\), also a central limit theorem is satisfied. If \(L(t)\ll t\), \(p^{L(t)}(0,t)\) decreases like the quenched survival probability. If \(t\ll L(t)\) and \(\log L(t)\ll t^{d/(d+2)}\) an intermediate regime is obtained.

Furthermore, when the dimension \(d=1\) it is possible to describe the fluctuations of \(p^{L(t)}(0,t)\) 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.

Reviewer: Farruh Mukhamedov (Aveiro)

##### MSC:

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

60J45 | Probabilistic potential theory |

60J65 | Brownian motion |

82C22 | Interacting particle systems in time-dependent statistical mechanics |

##### Keywords:

parabolic Anderson model; random walk; enlargement of obstacles; principal eigenvalue; Wiener sausage
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\textit{G. Ben Arous} et al., Ann. Probab. 33, No. 6, 2149--2187 (2005; Zbl 1099.82003)

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