Fluctuations of the free energy in the REM and the $p$-spin SK models

Abstract

We consider the random uctuations of the free energy in the p-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a r e c e n t paper by T alagrand on the p-spin version, we prove that (for p even) the random corrections to the free energy are on a scale N ;(p;2)=4 only, and after proper rescaling converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, , smaller than a critical p . We also show that p ! p 2 l n 2 a s p " +1. Additionally we study the formal p " +1 limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at all temperatures. For < p 2 l n 2 , uctuations are found at an exponentially small scale, with two distinct limit laws above and below a second critical value p ln 2=2: For up to that value the rescaled uctuations are Gaussian, while below that there are non-Gaussian uctuations driven by t h e P oisson process of the extreme values of the random energies. For larger than the critical p 2 l n 2 , the uctuations of the logarithm of the partition function are on scale one and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to 1=2.

Limit theorems for sums of random exponentials

“…Eisele [8] and Olivieri and Picco [13] have rigorously derived the limit (1.3) (in probability and a.s.) and also extended this result to the case where X i have the Weibull-type tail (1.2) (case B). 1 Recently, a detailed analysis of the limit laws for Z n (β) in the Gaussian case has been accomplished by Bovier et al [6]. In particular, is has been shown that in addition to the phase transition at the critical point β c , manifested as the LLN breakdown for β > β c , within the region β < β c there is a second phase transition atβ c = log 2/2 = 1 2 β c , in that for β >β c the fluctuations of Z n (β) become non-Gaussian.…”

“…In the present work, we extend these results to the class of distributions with Weibull/Fréchet-type tails of the form (1.2). As compared to the paper [6] which proceeded from extreme value theory, we use methods of theory of summation of independent random variables. This general and powerful approach simplifies the proofs and in particular reveals that non-Gaussian limit laws are in fact stable.…”

Phase transition in the Jarzynski estimator of free energy differences

“…In the appropriate scaling limit, the appearance of the bias is abrupt and corresponds to a phase transition in variants of the random energy model (REM) [20][21][22][23][24]. The connection between the random energy model and the Jarzynski estimator of free energy differences was first made in Refs.…”

Fluctuations of the partition function in the generalized random energy model with external field

“…The point process ͑1.18͒ is the limiting object which appears in the REM. 15 Indeed, in this case t ‫ء‬ ͑0͒ =0, and, hence, ͑0͒ = M͑0͒ = ͱ 2 log 2, which in turn implies that the scaling constants ͑1.15͒ and ͑1.16͒ coincide with that of the REM without external field.…”