REM universality and Poisson-Dirichlet Gibbs weights for linear random energy

Abstract

We study the Hamiltonian $H_n(h,σ)=\sum_{i=1}^n h_i(σ_i-m), $ where $(h_i)$ are i.i.d.\ real random variables and $(σ_i)$ are i.i.d.\ Ising spins. We consider the energy levels obtained after an independent thinning that retains an exponential number of configurations ($e^{O(n)}$). We prove that, after an $(h_i)$-dependent centering, the resulting point process converges in distribution to a Poisson point process with exponential intensity. Thus, the energy levels asymptotically has the one of the Random Energy Model (REM). Our results extend previous ones, where REM universality for this model was established only either for energy fluctuations of order $e^{-O(n)}$ or for $e^{o(\sqrt n)}$ randomly selected configurations. We also identify the limiting Gibbs weights, which converge to a Poisson--Dirichlet law, and the quenched free energy, which exhibits a freezing transition at $β=\tildeλ$. The proofs are presented here in compressed form; full details are given in the companion preprint.