Intermittency on catalysts

TL;DR

This paper reviews recent results on intermittency phenomena in the Parabolic Anderson Model within various time-dependent random media, highlighting how catalyst type, dimension, and diffusion influence growth rates.

Contribution

It provides an overview of new findings on how different catalysts affect intermittency and Lyapunov exponents in the Parabolic Anderson Model.

Findings

Lyapunov exponents depend on dimension and diffusion constant

Different catalysts lead to distinct intermittency behaviors

Results cover random walks, exclusion process, and voter model

Abstract

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.