1,723 research outputs found
Intermittency in a catalytic random medium
In this paper, we study intermittency for the parabolic Anderson equation
, where , is the diffusion constant, is the
discrete Laplacian and is a
space-time random medium. We focus on the case where is times
the random medium that is obtained by running independent simple random walks
with diffusion constant starting from a Poisson random field with
intensity . Throughout the paper, we assume that
. The solution of the equation describes
the evolution of a ``reactant'' under the influence of a ``catalyst''
. We consider the annealed Lyapunov exponents, that is, the exponential
growth rates of the successive moments of , and show that they display an
interesting dependence on the dimension and on the parameters
, with qualitatively different intermittency behavior
in , in and in . Special attention is given to the
asymptotics of these Lyapunov exponents for and .Comment: Published at http://dx.doi.org/10.1214/009117906000000467 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Intermittency on catalysts: three-dimensional simple symmetric exclusion
We continue our study of intermittency for the parabolic Anderson model
in a space-time random medium
, where is a positive diffusion constant, is the lattice
Laplacian on , , and is a simple symmetric exclusion
process on in Bernoulli equilibrium. This model describes the evolution
of a \emph{reactant} under the influence of a \emph{catalyst} .
In G\"artner, den Hollander and Maillard (2007) we investigated the behavior
of the annealed Lyapunov exponents, i.e., the exponential growth rates as
of the successive moments of the solution . This led to an
almost complete picture of intermittency as a function of and . In
the present paper we finish our study by focussing on the asymptotics of the
Lyaponov exponents as in the \emph{critical} dimension ,
which was left open in G\"artner, den Hollander and Maillard (2007) and which
is the most challenging. We show that, interestingly, this asymptotics is
characterized not only by a \emph{Green} term, as in , but also by a
\emph{polaron} term. The presence of the latter implies intermittency of
\emph{all} orders above a finite threshold for .Comment: 38 page
Intermittency on catalysts
The present paper provides an overview of results obtained in four recent
papers by the authors. These papers address the problem of intermittency for
the Parabolic Anderson Model in a \emph{time-dependent random medium},
describing the evolution of a ``reactant'' in the presence of a ``catalyst''.
Three examples of catalysts are considered: (1) independent simple random
walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is
on the annealed Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of the reactant. It turns out that these exponents exhibit
an interesting dependence on the dimension and on the diffusion constant.Comment: 11 pages, invited paper to appear in a Festschrift in honour of
Heinrich von Weizs\"acker, on the occasion of his 60th birthday, to be
published by Cambridge University Pres
Intermittency on catalysts: Voter model
In this paper we study intermittency for the parabolic Anderson equation
with
, where is
the diffusion constant, is the discrete Laplacian,
is the coupling constant, and
is a space--time random medium.
The solution of this equation describes the evolution of a ``reactant''
under the influence of a ``catalyst'' . We focus on the case where
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
or the equilibrium measure , where is the density of
1's. We consider the annealed Lyapunov exponents, that is, the exponential
growth rates of the successive moments of . We show that if the random walk
transition kernel has zero mean and finite variance, then these exponents are
trivial for , but display an interesting dependence on the
diffusion constant for , with qualitatively different
behavior in different dimensions. In earlier work we considered the case where
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.Comment: Published in at http://dx.doi.org/10.1214/10-AOP535 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
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