Intermittency on catalysts: three-dimensional simple symmetric exclusion

J. Gärtner, W.Th.F. Hollander, den, G. Maillard

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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.
Original languageEnglish
Pages (from-to)2091-2129
JournalElectronic Journal of Probability
Publication statusPublished - 2009


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