Perfect sampling of Ising spin glasses with bounding chains

Abstract

We study perfect sampling algorithms for Ising spin glasses based on coupling from the past and bounding chains, on several graph families and coupling distributions. For the single-site bounding chain, we establish coalescence in $O(n\log n)$ updates under Dobrushin conditions. On sparse random graphs, our numerical results suggest that efficient sampling extends throughout the tree-uniqueness regime, with a slowdown beyond this threshold. On Erdős-Rényi random graphs $\mathcal{G}(n,d/n)$ with standard Gaussian couplings, we introduce a block bounding chain and prove its coalescence in $O(n\log n)$ block updates in the tree-uniqueness regime for sufficiently large $d$, each block update requiring $O(n)$ operations, adapting a proof in~\cite{Efthymiou_Zampetakis_2024} for the rapid mixing of block Glauber dynamics. On periodic lattices, our experiments instead suggest that the single-site bounding chain is limited by the looseness of its bounding envelope, whereas on the Sherrington-Kirkpatrick model, slow coalescence is accompanied by damage spreading at every inverse temperature tested. These observations motivate alternative bounding representations and couplings as strategies to improve perfect sampling on spin glasses.