Intermittency on catalysts: three-dimensional simple symmetric exclusion

Abstract

We continue our study of intermittency for the parabolic Anderson model ∂u/∂t = κ∆u + ξu in a space-time random medium ξ, where κ is a positive diffusion constant, ∆ is the lattice Laplacian on Z d , d ≥ 1, and ξ is a simple symmetric exclusion process on Z d in Bernoulli equilibrium. This model describes the evolution of a reactant u under the influence of a catalyst ξ.In [3] we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as t → ∞ of the successive moments of the solution u. This led to an almost complete picture of intermittency as a function of d and κ. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as κ → ∞ 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 κ.MSC 2000. Primary 60H25, 82C44; Secondary 60F10, 35B40.

Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

“…The case where ξ is a single random walk was studied by Gärtner and Heydenreich [12]. Gärtner, den Hollander and Maillard [14], [16], [17] subsequently considered the cases where ξ is an exclusion process with an irreducible symmetric random walk transition kernel starting from a Bernoulli product measure (SEP), respectively, a voter model with an irreducible symmetric transient random walk transition kernel starting either from a Bernoulli product measure or from equilibrium (SVM). In each of these cases, a fairly complete picture of the behavior of the annealed Lyapunov exponents was obtained, including the presence or absence of intermittency, i.e., λ p (κ) > λ p−1 (κ) for some or all values of p ∈ N\{1} and κ ∈ [0, ∞).…”

“…We refer the reader to Gärtner, den Hollander and Maillard [15] for an overview. It was shown in Gärtner and den Hollander [13], and Gärtner, den Hollander and Maillard [14], [16], [17] that for ISRW, SEP and SVM in equilibrium the function κ → λ p (κ) satisfies:…”

The Parabolic Anderson Model

“…Interestingly, in d D 3, the asymptotics of Ä p .Ä/ show a remarkable connection with the polaron model, whose meanfield version we briefly mentioned in Example 7.11; there is a heuristics for deeper reasons behind it. Þ Example 8.5 (Symmetric exclusion process) Let us describe the results of [GärHolMai07] and [GärHolMai09a] for the model (ii). Þ Remark 8.4 (Survival and extinction of branching random walks with catalysts) The interpretation of interacting reactants and catalysts in the model (i) above has also been studied in [KesSid03] with the additional assumption that reactant particles die at a certain deterministic rate ı 2 .0; 1/.…”

Parabolic Anderson Model with a Finite Number of Moving Catalysts

“…Here, due to the lack of ergodicity, such a geometric picture of intermittency is not available. Nevertheless, (7) can still be interpreted as the p-th moment of u being generated by some exponentially rare event (see [3], Section 1.2 for a more detailed analysis).…”