From torpid to rapid mixing: group averaging for a weakly interacting Ising star
Abstract
We study lazy single-site Metropolis dynamics Pβ at inverse temperature β≥ 0 on {−1,+1}d for an Ising star with additional signed interactions among the leaves. If the absolute row sums of the leaf-interaction matrix are at most κ≤1/2, the worst-case total-variation mixing time of Pβ is at least of order dexp{cβ(d−1)} for β≥1, with universal c>0. Averaging over global spin reversal reduces the mixing time to 𝒪(d2(1+β)) for both GPβG and (Pβ+G)/2, where G is the Gibbs kernel induced by the partition of the state space into spin-reversal orbits {−x,x}. Partition-function interpolation gives the lower bound for Pβ. The upper bounds follow from Wu's Dobrushin inequality and a decomposition into projection and restriction chains. This gives an explicit example in which group averaging turns an exponentially slow chain into a polynomially fast one.
Related
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BibTeX
@misc{choi2026from,
title = {{From torpid to rapid mixing: group averaging for a weakly interacting Ising star}},
author = {Michael C. H. Choi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15023},
eprint = {2609.15023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15023},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From torpid to rapid mixing: group averaging for a weakly interacting Ising star
AU - Michael C. H. Choi
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15023
UR - https://arxiv.org/abs/2609.15023
AB - We study lazy single-site Metropolis dynamics P_β at inverse temperature β≥ 0 on {−1,+1}^d for an Ising star with additional signed interactions among the leaves. If the absolute row sums of the leaf-interaction matrix are at most κ≤1/2, the worst-case total-variation mixing time of P_β is at least of order dexp{cβ(d−1)} for β≥1, with universal c>0. Averaging over global spin reversal reduces the mixing time to 𝒪(d^2(1+β)) for both GP_βG and (P_β+G)/2, where G is the Gibbs kernel induced by the partition of the state space into spin-reversal orbits {−x,x}. Partition-function interpolation gives the lower bound for P_β. The upper bounds follow from Wu's Dobrushin inequality and a decomposition into projection and restriction chains. This gives an explicit example in which group averaging turns an exponentially slow chain into a polynomially fast one.
ER -