Finite Size Corrections to the Parisi Overlap Function in the GREM

Abstract

We investigate the effects of finite size corrections on the overlap probabilities in the Generalized Random Energy Model (GREM) in two situations where replica symmetry is broken in the thermodynamic limit. Our calculations do not use replicas, but shed some light on what the replica method should give for finite size corrections. In the gradual freezing situation, which is known to exhibit full replica symmetry breaking, we show that the finite size corrections lead to a modification of the simple relations between the sample averages of the overlaps Y k between k configurations predicted by replica theory. 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. We show that finite size corrections lead to full replica symmetry breaking and give a more complete derivation of the results presented in [2] for the directed polymer on a tree.

Possible instability of one-step replica symmetry breaking in p -spin Ising models outside mean-field theory

“…Billoire et al [14] studied P (q) of the p = 3 fully-connected model for values of N up to 192. Their P (q) looks remarkably similar to the finite size form predicted by Mottishaw and Derrida [15] for the generalized random energy model (GREM) when FRSB is present. Unfortunately the treatment in Ref.…”

“…(27) also reduces to the one for the 1RSB scheme, Eq. (15). We note that this solution is independent of m 1 .…”

“…Unfortunately the treatment in Ref. [15] was not done using replicas so it is not obvious how it might be extended directly to the finite N fully-connected p-spin model.…”

Spin Glass Theory and Far Beyond

“…For a GREM with an arbitrary number k of levels, if all the α i and all the a i are equal, the extensive part of the free energy and the overlaps are still given by (1.8,1.18,1.19). The finite size corrections are however different [40,27,71]. The same is true [43] for the directed polymer on a binary tree, of height N , where on the 2 N +1 − 2 edges b there are i.i.d.…”