Stratified Sampling for the Ising Model: A Graph-Theoretic Approach

TL;DR

This paper introduces a graph-theoretic sampling method to estimate the partition function and thermodynamic properties of the ferromagnetic Ising model, offering a practical alternative to Markov chain Monte Carlo techniques.

Contribution

It proposes a novel approach combining graph theory and heuristic sampling to compute temperature-independent coefficients for the Ising model.

Findings

Efficient estimation of the partition function.

Accurate computation of thermodynamic quantities.

Potential for practical application in statistical physics.

Abstract

We present a new approach to a classical problem in statistical physics: estimating the partition function and other thermodynamic quantities of the ferromagnetic Ising model. Markov chain Monte Carlo methods for this problem have been well-studied, although an algorithm that is truly practical remains elusive. Our approach takes advantage of the fact that, for a fixed bond strength, studying the ferromagnetic Ising model is a question of counting particular subgraphs of a given graph. We combine graph theory and heuristic sampling to determine coefficients that are independent of temperature and that, once obtained, can be used to determine the partition function and to compute physical quantities such as mean energy, mean magnetic moment, specific heat, and magnetic susceptibility.