Finite size corrections to the Parisi overlap function in the GREM

TL;DR

This paper analyzes finite size effects on overlap probabilities in the GREM, revealing how corrections modify replica symmetry breaking predictions and induce fluctuations, with implications for understanding spin glass models.

Contribution

It provides a non-replica method analysis of finite size corrections in GREM, showing how these corrections alter replica symmetry breaking scenarios.

Findings

Finite size corrections modify overlap relations in full RSB regimes.

Corrections induce fluctuations with negative variance in replica block sizes.

Finite size effects can lead to full RSB in one-step RSB models.

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 \emph{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.