Fluctuations of the partition function in the GREM with external field

TL;DR

This paper analyzes the fluctuations of the partition function in the GREM with external field, revealing a hierarchical structure and providing explicit formulas for free energy and large deviations.

Contribution

It introduces a hierarchical coarse-graining approach to describe fluctuations in the GREM with external field and derives explicit free energy formulas.

Findings

Fluctuations follow a hierarchical structure after coarse-graining.

Explicit formula for the free energy of the GREM with external field.

Identification of an optimal magnetization in the thermodynamic limit.

Abstract

We study Derrida's generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by strength of external field and by parameters of the GREM.