Beyond Windability: An FPRAS for The Six-Vertex Model

TL;DR

This paper introduces a new FPRAS for the six-vertex model that works with unwindable constraint functions using MCMC and Glauber dynamics, expanding the applicability of approximation algorithms in statistical physics.

Contribution

It presents the first FPRAS for the six-vertex model with unwindable constraints, employing Glauber dynamics and circuit decomposition for rapid mixing analysis.

Findings

Developed an FPRAS for unwindable six-vertex models

Used Glauber dynamics with circuit decomposition for Markov Chain design

Proved rapid mixing via coupling method

Abstract

The six-vertex model is an important model in statistical physics and has deep connections with counting problems. There have been some fully polynomial randomized approximation schemes (FPRAS) for the six-vertex model [30, 10], which all require that the constraint functions are windable. In the present paper, we give an FPRAS for the six-vertex model with an unwindable constraint function by Markov Chain Monte Carlo method (MCMC). Different from [10], we use the Glauber dynamics to design the Markov Chain depending on a circuit decomposition of the underlying graph. Moreover, we prove the rapid mixing of the Markov Chain by coupling, instead of canonical paths in [10].