🤖 AI Summary
This work addresses the overly restrictive weight conditions in approximating partition functions of abstract polymer models, which severely limit algorithmic applicability. We propose a dynamic sampling framework based on incompatible polymer clusters. Our approach is the first to algorithmically apply polymer models to continuous hard-sphere mixtures, significantly broadening the parameter regimes where approximation algorithms—including Markov chain Monte Carlo (MCMC) and cluster expansions—remain valid, via a relaxed cluster dynamics condition. Technically, we integrate cluster-driven Markov chains, α-expander graph theory, discretization approximations, and hardcore model analysis. Theoretically, we establish that for hardcore models on bipartite α-expander graphs, the admissible parameter range improves by at least a factor of $2^{1/alpha}$; moreover, we provide the first rigorous polynomial-time approximation algorithm for the partition function of hard-sphere mixtures.
📝 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. The clique dynamics condition implies rapid mixing of a Markov chain that utilizes 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. Additionally, we apply our method to approximate the partition function of the multi-component hard-sphere model, a continuous model of spherical particles in the Euclidean space. To this end, we define a discretization that allows us to bound the rate of convergence to the continuous model. To the best of our knowledge, this is the first algorithmic application of polymer models to a continuous geometric problem and the first rigorous computational result for hard-sphere mixtures.
