Computational thresholds for the fixed-magnetization Ising model

DOI10.1145/3519935.3520003arXiv2111.03033OpenAlexW3209539671MaRDI QIDQ6083601

Alexandra Kolla, Will Perkins, Ewan Davies, Unnamed Author

Publication date: 8 December 2023

Published in: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (Search for Journal in Brave)

Abstract: The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferromagnetic Ising model with uniform external field is tractable at all temperatures and on all graphs, due to the randomized algorithm of Jerrum and Sinclair. Here we show that hidden inside the model are hard computational problems. For the class of bounded-degree graphs we find computational thresholds for the approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization (that is, fixing the number of $+1$ and $-1$ spins). In particular, letting $$ denote the critical inverse temperature of the zero-field Ising model on the infinite $Delta$-regular tree, and $$ denote the mean magnetization of the zero-field $+$ measure on the infinite $Delta$-regular tree at inverse temperature $$, we prove, for the class of graphs of maximum degree $Delta$: 1. For $$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $eta$. 2. For $$, there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $eta$ such that $$. 3. For $$, there is no FPRAS for the fixed-magnetization Ising model for magnetizations $eta$ such that $$ unless NP=RP@.

Full work available at URL: https://arxiv.org/abs/2111.03033

zbMATH Keywords

Ising modellocal central limit theoremfixed magnetizationapproximate counting and samplingcomputational threshold

Mathematics Subject Classification ID

Theory of computing (68Qxx)

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