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Intermittency on catalysts: three-dimensional simple symmetric exclusion
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Jürgen Gärtner, Institut für Mathematik, Technische Universität Berlin
Frank den Hollander, Mathematical Institute, Leiden University
Grégory Maillard, CMI-LATP, Université de Provence
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Abstract
We continue our study of intermittency for the parabolic Anderson model,
∂u ⁄ ∂ t = κΔu + ξu in a space-time random
medium ξ, where κ is a positive diffusion constant, Δ is
the lattice Laplacian on Zd, d≥1, and ξ 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" ξ.
In Gärtner, den Hollander and Maillard [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 Lyapunov 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.
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Full text: PDF
Pages: 2091-2129
Published on: September 28, 2009
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Electronic Journal of Probability. ISSN: 1083-6489 |
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