Finite size corrections in the random energy model and the replica approach

Abstract

We present a systematict and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as "complex replica numbers". Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the "complex replica numbers" near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.

Finite Size Corrections to the Parisi Overlap Function in the GREM

“…This can be interpreted as fluctuations in the replica block size with a negative variance. The mechanism is similar to the one we found recently in the random energy model [1]. We also consider a simultaneous freezing situation, which is known to exhibit one step replica symmetry breaking.…”

“…In this paper we will study a variant of the original GREM that we will refer to as the Poisson GREM. As in the REM [1] this has the advantage of simplifying some of the analysis while giving the same results for large systems, including the leading finite size corrections (see appendix B).…”

“…In order to shed some light on this problem, we have investigated some exactly soluble models for which the Parisi overlap function ( [7,8,9] and references therein) can be calculated without using the replica method. This was done in two recent works for the random energy model (REM) [1] and for the problem of the directed polymer on a tree [2]. Both models exhibit a one step broken symmetry of replica [10,11,12].…”

“…Y k (q) is defined to be the probability that k replicas with the same quenched disorder have an overlap of at least q. If q C,C ′ is the overlap between a pair of configurations and E C is the energy of a configuration, then Y k (q) is therefore defined by Y k (q) = C1 · · · C k 1≤i<j≤k Θ(q Ci,Cj − q) e −β(EC 1 +···+EC k ) Z k (1) where Z is the partition function (Z = C e −βEC ). (According to the theory of mean field spin glasses the Y k (q) are in general sample dependent and have sample to sample fluctuations of order 1 with statistics predicted by the theory [8,18].…”

“…In terms of Parisi's replica symmetry breaking scheme (see section 3) this means finite size effects can be interpreted as fluctuations in the replica block size and as in the REM case [1], with a negative variance of these fluctuations.…”