Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

TL;DR

This paper investigates the long-term behavior of solutions to the parabolic Anderson model with doubly-exponential tails, revealing mass concentration near optimal sites and establishing aging phenomena through eigenvalue analysis.

Contribution

It introduces a detailed analysis of mass concentration and aging in the parabolic Anderson model with heavy-tailed randomness, utilizing eigenvalue order statistics.

Findings

Mass concentrates near sites balancing eigenvalues and distance.

Processes of site location and mass growth converge under scaling.

Aging phenomena are rigorously established for the model.

Abstract

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb Z^d$ and $\xi=(\xi(z))_{z\in\mathbb Z^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $\Delta+\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto \tfrac1t \log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $\Delta+\xi$ in large sets recently proved by the first two authors.