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Cycle Basis Markov Chains for the Ising Model



Noah S. Streib, Amanda A. Streib


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


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


Streib, N. and Streib, A. (2017), Cycle Basis Markov Chains for the Ising Model, ACM-SIAM Symposium on Discrete Algorithms, Portland, OR, US, [online],, (Accessed July 16, 2024)


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Created December 31, 2016, Updated October 12, 2021