On a nonhierarchical generalization of the Perceptron GREM

TL;DR

This paper introduces a nonlinear, nonhierarchical extension of Derrida's GREM, deriving a Parisi formula for the limiting free energy using large deviation analysis, aligning with Parisi theory predictions.

Contribution

It presents a novel nonhierarchical generalization of GREM and establishes a Parisi formula for its free energy through rigorous large deviation methods.

Findings

Derived a Parisi formula for the new model's free energy

Established a Boltzmann-Gibbs principle for the model

Confirmed the model's free energy aligns with Parisi theory predictions

Abstract

We introduce a nonlinear, nonhierarchical generalization of Derrida's GREM and establish through a Sanov-type large deviation analysis both a Boltzmann-Gibbs principle as well as a Parisi formula for the limiting free energy. In line with the predictions of the Parisi theory, the free energy is given by the minimal value over all Parisi functionals/hierarchical structures in which the original model can be coarse-grained.