Delocalising the parabolic Anderson model through partial duplication of the potential

TL;DR

This paper studies a modified parabolic Anderson model with partially duplicated potential, revealing a phase transition between localization and delocalization depending on the tail decay of the potential distribution.

Contribution

It introduces a new variant of the model with partial potential duplication and demonstrates a phase transition influenced by the tail decay exponent.

Findings

For large decay exponents, the model localizes similarly to the i.i.d. case.

For small decay exponents, the model can delocalize with mass on multiple sites.

A phase transition occurs depending on the tail decay rate of the potential.

Abstract

The parabolic Anderson model on $\mathbb{Z}^d$ with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small values of the exponent we show that the model may delocalise. More precisely, we show that there is an event of non-negligible probability on which the solution has non-negligible mass on two sites.