The parabolic Anderson model on the hypercube

TL;DR

This paper studies the parabolic Anderson model on the hypercube with random potential, revealing a phase transition in growth behavior over time and connecting it to population genetics and the Random Energy Model.

Contribution

It provides a detailed analysis of the phase transition in the model's growth dynamics and its implications for mutation-selection processes on random fitness landscapes.

Findings

Short time growth resembles a system without diffusion.

Long time growth is governed by principal eigenvalues.

Transition time depends on potential differences.

Abstract

We consider the parabolic Anderson model $\frac{\partial}{\partial t} v_n=\kappa\Delta_n v_n + \xi_n v_n$ on the $n$-dimensional hypercube $\{-1,+1\}^n$ with random i.i.d. potential $\xi_n$. We parametrize time by volume and study $v_n$ at the location of the $k$-th largest potential, $x_{k,2^n}$. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales $v_n(t_n,x_{k,2^n})$ behaves like a system without diffusion and grows as $\exp\big\{(\xi_n(x_{k,2^n}) - \kappa)t_n\big\}$, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the operator $\kappa \Delta_n+\xi_n$, for which we give precise asymptotics. Moreover, the transition time depends only on the difference $\xi_n(x_{1,2^n})-\xi_n(x_{k,2^n})$. One of our main motivations in this article is to investigate the mutation-selection model of population genetics on a random fitness landscape, which is given by the ratio of $v_n$ to its total mass, with $\xi_n$ corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at $x_{k,2^n}$ moves completely to $x_{1,2^n}$ on time scales where the transition of growth rates happens. The class of potentials we consider involves the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.