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The parabolic Anderson model on the hypercube

We consider the parabolic Anderson model $\frac{\partial}{\partial t} v_n=κΔ_n v_n + ξ_n v_n$ on the $n$-dimensional hypercube $\{-1,+1\}^n$ with random i.i.d. potential $ξ_n$. 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 $\exp\big\{(ξ_n(x_{k,2^n}) - κ)t_n\big\}$, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the operator $κΔ_n+ξ_n$, for which we give precise asymptotics. Moreover, the transition time depends only on the difference $ξ_n(x_{1,2^n})-ξ_n(x_{k,2^n})$. 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 $ξ_n$ 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.