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 and (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, where the weight statistics coincides with the weight statistics of Lévy sums with index mu=TT{c}<1 . In this paper, we study the weight statistics at criticality via the entropy S=Sigma w{i}lnw{i} and the generalized moments Y{k}= Sigma 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 S[over]{N}(T{c}) approximately N{12} , the typical values e{lnY{k}[over]} decay as N{-k2} , and the disorder-averaged values Y{k}[over] are governed by rare events and decay as N{-12} for any k>1 . For the DPCT, we find that the entropy scales similarly as S[over]{N}(T{c}) approximately N{12} , whereas another exponent nu'=1 governs the Y{k} statistics: the typical values e{lnY{k}[over]} decay as N{-k} , and the disorder-averaged values Y{k}[over] decay as N{-1} for any k>1 . As a consequence, the asymptotic probability distribution pi[over]{N=infinity}(q) of the overlap q , in addition to 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-TT{c})delta(q-1) of the low-temperature phase T<T{c} . The associated value pi[over]{N=infinity}(q=1) is finite for the DPCT, and diverges as pi[over]{N=infinity}(q=1) approximately N{12} for the REM.