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September 2010 Intermittency on catalysts: Voter model
J. Gärtner, F. den Hollander, G. Maillard
Ann. Probab. 38(5): 2066-2102 (September 2010). DOI: 10.1214/10-AOP535


In this paper we study intermittency for the parabolic Anderson equation ∂u/∂t=κΔu+γξu with u : ℤd×[0, ∞)→ℝ, where κ∈[0, ∞) is the diffusion constant, Δ is the discrete Laplacian, γ∈(0, ∞) is the coupling constant, and ξ : ℤd×[0, ∞)→ℝ is a space–time random medium. The solution of this equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ.

We focus on the case where ξ is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure νρ or the equilibrium measure μρ, where ρ∈(0, 1) is the density of 1’s. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of u. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for 1≤d≤4, but display an interesting dependence on the diffusion constant κ for d≥5, with qualitatively different behavior in different dimensions.

In earlier work we considered the case where ξ is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.


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J. Gärtner. F. den Hollander. G. Maillard. "Intermittency on catalysts: Voter model." Ann. Probab. 38 (5) 2066 - 2102, September 2010.


Published: September 2010
First available in Project Euclid: 17 August 2010

zbMATH: 1223.60080
MathSciNet: MR2722795
Digital Object Identifier: 10.1214/10-AOP535

Primary: 60H25 , 82C44
Secondary: 35B40 , 60F10

Keywords: catalytic random medium , Coalescing random walks , Intermittency , large deviations , Lyapunov exponents , Parabolic Anderson equation , voter model

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 5 • September 2010
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