The discrete random energy model and one step replica symmetry breaking

TL;DR

This paper analyzes a discrete version of the random energy model, revealing size-dependent oscillations in overlaps and proposing a novel replica approach with fluctuating block sizes to address these challenges.

Contribution

It introduces a solution for the discrete random energy model and proposes a modified replica method with complex block sizes to handle non-trivial overlaps.

Findings

Average overlaps oscillate with system size in the discrete model.

Standard replica symmetry breaking captures free energy but not overlaps.

Fluctuating complex block sizes offer a potential solution for overlap analysis.

Abstract

We solve the random energy model when the energies of the configurations take only integer values. In the thermodynamic limit, the average overlaps remain size dependent and oscillate as the system size increases. While the extensive part of the free energy can still be obtained by a standard replica calculation with one step replica symmetry breaking, it is no longer possible to recover the overlaps in this way. A possible way to adapt the replica approach is to allow the sizes of the blocks in the Parisi matrix to fluctuate and to take complex values.