A new phase transition in the parabolic Anderson model with partially duplicated potential

TL;DR

This paper studies a variant of the parabolic Anderson model with a partially duplicated potential, revealing a new phase transition depending on the duplication probability's relation to the distance from the origin.

Contribution

It introduces a new phase transition in the model when the duplication probability varies with distance, extending previous fixed-probability results.

Findings

Identifies a critical scale for duplication probability where the localization behavior changes.

Shows complete localization when duplication probability is below the critical scale.

Demonstrates localization on two sites when duplication probability exceeds the critical scale.

Abstract

We investigate a variant of the parabolic Anderson model, introduced in previous work, in which an i.i.d.\! potential is partially duplicated in a symmetric way about the origin, with each potential value duplicated independently with a certain probability. In previous work we established a phase transition for this model on the integers in the case of Pareto distributed potential with parameter $\alpha > 1$ and fixed duplication probability $p \in (0, 1)$: if $\alpha \ge 2$ the model completely localises, whereas if $\alpha \in (1, 2)$ the model may localise on two sites. In this paper we prove a new phase transition in the case that $\alpha \ge 2$ is fixed but the duplication probability $p(n)$ varies with the distance from the origin. We identify a critical scale $p(n) \to 1$, depending on $\alpha$, below which the model completely localises and above which the model localises on exactly two sites. We further establish the behaviour of the model in the critical regime.