Cycle Basis Markov Chains for the Ising Model

Published

January 1, 2017

Author(s)

Noah S. Streib, Amanda A. Streib

Abstract

The ferromagnetic Ising model with zero applied field reduces to sampling even subgraphs X of a graph G with probability proportional to $\lambda^|E(X)|}$, for $\lambda \in [0,1]$. In this paper, we present a class of Markov chains for sampling subgraphs with a fixed set of odd-degree vertices, which generalizes the classical single-site dynamics; when this set is empty, we sample even subgraphs. We use the fact that the set of even subgraphs of a graph $G$ forms a vector space generated by a cycle basis of $G$. In particular, we introduce Markov chains $\m(\C)$ whose transitions are defined by symmetric difference with elements of a cycle basis $\C$. We show that for any graph $G$ and any ''long'' cycle basis $\C$ of $G$, there is a $\lambda$ for which $\m(\C)$ requires exponential time to mix. All fundamental cycle bases of the $d$-dimensional grid are long. For the 2-dimensional grid with periodic boundary conditions we show that there is a $\lambda$ for which $\m(\C)$ requires exponential time to mix for \emphall} bases $\C$.

Conference Dates

January 5-7, 2014

Conference Location

Portland, OR, US

Conference Title

ACM-SIAM Symposium on Discrete Algorithms

Pub Type

Conferences

Download Paper

https://doi.org/10.1137/1.9781611974775.5

Local Download

Keywords

Ising, Markov chain, mixing, cycle basis, graph theory

Citation

Streib, N. and Streib, A. (2017), Cycle Basis Markov Chains for the Ising Model, ACM-SIAM Symposium on Discrete Algorithms, Portland, OR, US, [online], https://doi.org/10.1137/1.9781611974775.5, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=914152 (Accessed October 14, 2025)