TY - JOUR

T1 - Intermittency on catalysts: three-dimensional simple symmetric exclusion

AU - Gärtner, J.

AU - Hollander, den, W.Th.F.

AU - Maillard, G.

PY - 2009

Y1 - 2009

N2 - We continue our study of intermittency for the parabolic Anderson model ¿ u/¿ t = k ¿u+¿u in a space-time random medium ¿, where k is a positive diffusion constant, ¿ is the lattice Laplacian on Zd , d = 1, and ¿ is a simple symmetric exclusion process on Zd in Bernoulli equilibrium. This model describes the evolution of a reactant u under the influence of a catalyst ¿. In [3] we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as t ¿ 8 of the successive moments of the solution u. This led to an almost complete picture of intermittency as a function of d and k. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as k ¿ 8 in the critical dimension d = 3, which was left open in [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in d = 4, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for k.

AB - We continue our study of intermittency for the parabolic Anderson model ¿ u/¿ t = k ¿u+¿u in a space-time random medium ¿, where k is a positive diffusion constant, ¿ is the lattice Laplacian on Zd , d = 1, and ¿ is a simple symmetric exclusion process on Zd in Bernoulli equilibrium. This model describes the evolution of a reactant u under the influence of a catalyst ¿. In [3] we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as t ¿ 8 of the successive moments of the solution u. This led to an almost complete picture of intermittency as a function of d and k. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as k ¿ 8 in the critical dimension d = 3, which was left open in [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in d = 4, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for k.

M3 - Article

VL - 14

SP - 2091

EP - 2129

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -