Computational thresholds for the fixed-magnetization Ising model

TL;DR

This paper identifies computational thresholds for approximate counting and sampling in the fixed-magnetization ferromagnetic Ising model on bounded-degree graphs, revealing phase transitions in computational complexity.

Contribution

It establishes precise thresholds for when efficient algorithms exist or are unlikely for fixed-magnetization Ising models based on temperature and magnetization.

Findings

Efficient algorithms exist below the critical temperature for all magnetizations.

Above the critical temperature, algorithms work for large magnetizations.

Below the critical temperature, certain magnetizations are computationally hard.

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 $\beta_c(\Delta)$ denote the critical inverse temperature of the zero-field Ising model on the infinite $\Delta$-regular tree, and $\eta_{\Delta,\beta,1}^+$ denote the mean magnetization of the zero-field $+$ measure on the infinite $\Delta$-regular tree at inverse temperature $\beta$, we prove, for the class of graphs of maximum degree $\Delta$: 1. For $\beta < \beta_c(\Delta)$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $\eta$. 2. For $\beta > \beta_c(\Delta)$, there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $\eta$ such that $|\eta| >\eta_{\Delta,\beta,1}^+ $. 3. For $\beta > \beta_c(\Delta)$, there is no FPRAS for the fixed-magnetization Ising model for magnetizations $\eta$ such that $|\eta| <\eta_{\Delta,\beta,1}^+ $ unless NP=RP\@.