Variational Bounds for the Generalized Random Energy Model

TL;DR

This paper derives variational bounds for the pressure of the REM and GREM models, extending existing methods with new proofs for the upper bounds and using probabilistic and convexity techniques.

Contribution

It introduces a novel proof for the upper bounds of the pressure in the GREM, generalizes Guerra's bounds, and identifies the random probability cascade as the key structure.

Findings

Established variational upper and lower bounds for REM and GREM pressure.

Generalized Guerra's broken replica symmetry bounds for the models.

Identified the random probability cascade as the appropriate overlap structure.

Abstract

We compute the pressure of the random energy model (REM) and generalized random energy model(GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra's ``broken replica symmetry bounds",and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand's concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.