Spin glasses on Bethe Lattices for large coordination number

TL;DR

This paper investigates spin glasses on random lattices with finite connectivity, using a high connectivity expansion and replica symmetry breaking, revealing that divergences in free energy corrections are artifacts of approximation levels.

Contribution

It introduces a two-step replica symmetry breaking approach to improve the high connectivity expansion for spin glasses on Bethe lattices, reducing divergence issues.

Findings

Divergences in free energy corrections decrease with more accurate replica symmetry breaking.

The 1/z expansion remains well-defined at zero temperature.

Higher replica symmetry breaking steps mitigate divergence artifacts.

Abstract

We study spin glasses on random lattices with finite connectivity. In the infinite connectivity limit they reduce to the Sherrington Kirkpatrick model. In this paper we investigate the expansion around the high connectivity limit. Within the replica symmetry breaking scheme at two steps, we compute the free energy at the first order in the expansion in inverse powers of the average connectivity (z), both for the fixed connectivity and for the fluctuating connectivity random lattices. It is well known that the coefficient of the 1/z correction for the free energy is divergent at low temperatures if computed in the one step approximation. We find that this annoying divergence becomes much smaller if computed in the framework of the more accurate two steps breaking. Comparing the temperature dependance of the coefficients of this divergence in the replica symmetric, one step and two steps replica symmetry breaking, we conclude that this divergence is an artefact due to the use of a finite number of steps of replica symmetry breaking. The 1/z expansion is well defined also in the zero temperature limit.