On sampling two spin models using the local connective constant

TL;DR

This paper derives new optimal mixing bounds for Glauber dynamics on the Hard-core and Ising models, using the local connective constant of the graph, improving existing bounds and employing advanced spectral methods.

Contribution

It introduces novel mixing bounds based on the local connective constant, extending previous results and utilizing the $k$-non-backtracking matrix in spectral independence analysis.

Findings

Bounds include max-degree as a special case

Improved running time for FPTAS on general graphs

Enhanced mixing bounds using spectral radius and non-backtracking matrices

Abstract

This work establishes novel optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models. These bounds are expressed in terms of the local connective constant of the underlying graph $G$. This is a notion of effective degree for $G$. Our results have some interesting consequences for bounded degree graphs: (a) They include the max-degree bounds as a special case (b) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, \v Stefankoni\v c and Yin: PTRF 2017] for general graphs (c) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on [Hayes: FOCS 2006]. We obtain our results using tools from the theory of high-dimensional expanders and, in particular, the Spectral Independence method [Anari, Liu, Oveis-Gharan: FOCS 2020]. We explore a new direction by utilising the notion of the $k$-non-backtracking matrix $H_{G,k}$ in our analysis with the Spectral Independence. The results with $H_{G,k}$ are interesting in their own right.