The parabolic Anderson model on the hypercube

We consider the parabolic Anderson model $\frac{\partial}{\partial t}v_n=\kappa \Delta_n v_n +\xi_nv_n$ on the n-dimensional hypercube $\{-1,+1\}^n$ with random i.i.d. potential $\xi_n$. We study $v_n$ at the location of the kth 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 $v_m(t_n,x_{k,2^n})$ behaves like a system without diffusion and grows as $\exp{(\xi_n(x_{k,2^n})-\kappa)t_n}$, whereas, for long time scales the growth is dictated by the principal 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 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 above mentioned phase transition translates to the mutation–selection model as follows: a population initially concentrated at $\xi_n(x_{k,2^n})$ moves completely to $\xi_n(x_{1,2^n})$ on time scales where the transition of growth rates occurs. The class of potentials we consider includes the Random Energy Model (REM) which is studied in the statistical physics literature as one of the main examples of a random fitness landscape.