On the critical weight statistics of the Random Energy Model and of the Directed Polymer on the Cayley Tree

Abstract

We consider the critical point of two mean-field disordered models : (i) the Random Energy Model (REM), introduced by Derrida as a mean-field spin-glass model of $N$ spins (ii) the Directed Polymer of length $N$ on a Cayley Tree (DPCT) with random bond energies. Both models are known to exhibit a freezing transition between a high temperature phase where the entropy is extensive and a low-temperature phase of finite entropy. In this paper, we study the weight statistics at criticality via the entropy $S=-\sum w_i \ln w_i$ and the generalized moments $Y_k=\sum w_i^k$, where the $w_i$ are the Boltzmann weights of the $2^N$ configurations. In the REM, we find that the critical weight statistics is governed by the finite-size exponent $\nu=2$ : the entropy scales as $\bar{S}_N(T_c) \sim N^{1/2}$, the typical values $e^{\bar{\ln Y_k}}$ decay as $N^{-k/2}$, and the disorder-averaged values $\bar{Y_k}$ are governed by rare events and decay as $N^{-1/2}$ for any $k>1$. For the DPCT, we find that the entropy scales similarly as $\bar{S}_N(T_c) \sim N^{1/2}$, whereas another exponent $\nu'=1$ governs the $Y_k$ statistics : the typical values $e^{\bar{\ln Y_k}}$ decay as $N^{-k}$, the disorder-averaged values $\bar{Y_k}$ decay as $N^{-1}$ for any $k>1$. As a consequence, the asymptotic probability distribution $\bar{\pi}_{N=\infty}(q)$ of the overlap $q$, beside the delta function $\delta(q)$ which bears the whole normalization, contains an isolated point at $q=1$, as a memory of the delta peak $(1-T/T_c) \delta(q-1)$ of the low-temperature phase $T<T_c$. The associated value $\bar{\pi}_{N=\infty}(q=1)$ is finite for the DPCT, and diverges as $\bar{\pi}_{N=\infty}(q=1) \sim N^{1/2}$ for the REM.