Polymer Dynamics via Cliques: New Conditions for Approximations

TL;DR

This paper introduces the clique dynamics condition, a new less restrictive criterion for polymer models, enabling improved approximation algorithms for partition functions in statistical physics and combinatorics.

Contribution

It establishes the clique dynamics condition and demonstrates its use in designing faster mixing Markov chains for polymer models, broadening applicable parameter ranges.

Findings

The clique dynamics condition is less restrictive than existing conditions.

The new Markov chain achieves rapid mixing under the clique dynamics condition.

Improved approximation bounds for the hard-core model on bipartite expanders.

Abstract

Abstract polymer models are systems of weighted objects, called polymers, equipped with an incompatibility relation. An important quantity associated with such models is the partition function, which is the weighted sum over all sets of compatible polymers. Various approximation problems reduce to approximating the partition function of a polymer model. Central to the existence of such approximation algorithms are weight conditions of the respective polymer model. Such conditions are derived either via complex analysis or via probabilistic arguments. We follow the latter path and establish a new condition -- the clique dynamics condition -- , which is less restrictive than the ones in the literature. We introduce a new Markov chain where the clique dynamics condition implies rapid mixing by utilizing cliques of incompatible polymers that naturally arise from the translation of algorithmic problems into polymer models. This leads to improved parameter ranges for several approximation algorithms, such as a factor of at least $2^{1/\alpha}$ for the hard-core model on bipartite $\alpha$-expanders.