Finite Size Effects for the Ising Model on Random Graphs with Varying Dilution

TL;DR

This paper studies how finite system size affects the equilibrium magnetization of an Ising model on random graphs with varying connectivity, using theoretical methods and simulations to analyze phase transitions.

Contribution

It introduces a combined replica and cavity approach to analyze finite size effects on the Ising model on diluted random graphs, with validation against numerical simulations.

Findings

Cavity method aligns better with numerical results.

Finite size corrections depend on the graph's dilution parameter.

Phase transition remains mean-field type across different dilutions.

Abstract

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with $N$ nodes and $N^{\gamma}$ edges, with $1 < \gamma \leq 2$. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of $\gamma$ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter $\gamma$, using two different approaches: a replica-based finite $N$ expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics.