Finite size corrections to disordered systems on Erd\"{o}s-R\'enyi random graphs

TL;DR

This paper derives analytical formulas for finite size corrections to free energy in disordered spin systems on sparse Erdős-Rényi graphs, validated through numerical experiments on the Random Field Ising Model.

Contribution

It provides explicit $O(1/N)$ correction formulas using replica and cavity methods, advancing understanding of finite size effects in disordered systems on sparse graphs.

Findings

Derived analytical expressions for finite size corrections in the replica symmetric phase.

Validated formulas through numerical simulations on the RFIM at zero temperature.

Showed corrections are linear combinations of free energies of open and closed chains.

Abstract

We study the finite size corrections to the free energy density in disorder spin systems on sparse random graphs, using both replica theory and cavity method. We derive an analytical expressions for the $O(1/N)$ corrections in the replica symmetric phase as a linear combination of the free energies of open and closed chains. We perform a numerical check of the formulae on the Random Field Ising Model at zero temperature, by computing finite size corrections to the ground state energy density.