Structure of finite-RSB asymptotic Gibbs measures in the diluted spin glass models

TL;DR

This paper proposes a method to support the Mézard-Parisi formula in diluted spin glass models by showing finite-RSB measures satisfy the ansatz after a Hamiltonian modification, advancing understanding of spin distributions.

Contribution

It demonstrates that a small Hamiltonian modification ensures finite-RSB measures follow the Mézard-Parisi ansatz, a step toward proving the formula for diluted spin glasses.

Findings

Finite-RSB measures satisfy the Mézard-Parisi ansatz after Hamiltonian modification.

A pathway is suggested to extend results to full-RSB measures.

Supports the validity of the Mézard-Parisi formula in diluted models.

Abstract

We suggest a possible approach to proving the M\'ezard-Parisi formula for the free energy in the diluted spin glass models, such as diluted K-spin or random K-sat model at any positive temperature. In the main contribution of the paper, we show that a certain small modification of the Hamiltonian in any of these models forces all finite-RSB asymptotic Gibbs measures in the sense of the overlaps to satisfy the M\'ezard-Parisi ansatz for the distribution of spins. Unfortunately, what is still missing is a description of the general full-RSB asymptotic Gibbs measures. If one could show that the general case can be approximated by finite-RSB case in the right sense then one could a posteriori remove the small modification of the Hamiltonian to recover the M\'ezard-Parisi formula for the original model.