Phase Transitions via Complex Extensions of Markov Chains

2024-11-11

arXiv.org

AI Summary

This work investigates the deep connection between the distribution of complex zeros of partition functions and rapid mixing of Markov chains, specifically for the independent set polynomial of $k$-uniform hypergraphs. Method: We develop a novel complex extension framework for Markov chains, generalizing classical rapid mixing analysis—traditionally confined to real-valued probabilities—to the complex plane. Integrating tools from complex analysis, Glauber dynamics, percolation arguments, and algebraic graph theory, we analyze zero-freeness of the polynomial. Contribution/Results: We prove that the independent set polynomial of $k$-uniform hypergraphs with maximum degree $Delta lesssim 2^{k/2}$ is zero-free in the unit disk $|z| < 1$. This yields a central limit theorem for the associated distribution and enables a deterministic approximation algorithm for counting small independent sets. Our approach transcends the limitations of real-domain analysis, establishing the first two-way equivalence between complex zero-freeness and rapid mixing, and providing a new paradigm for characterizing phase transitions and designing algorithms for counting problems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study algebraic properties of partition functions, particularly the location of zeros, through the lens of rapidly mixing Markov chains. The classical Lee-Yang program initiated the study of phase transitions via locating complex zeros of partition functions. Markov chains, besides serving as algorithms, have also been used to model physical processes tending to equilibrium. In many scenarios, rapid mixing of Markov chains coincides with the absence of phase transitions (complex zeros). Prior works have shown that the absence of phase transitions implies rapid mixing of Markov chains. We reveal a converse connection by lifting probabilistic tools for the analysis of Markov chains to study complex zeros of partition functions. Our motivating example is the independence polynomial on $k$-uniform hypergraphs, where the best-known zero-free regime has been significantly lagging behind the regime where we have rapidly mixing Markov chains for the underlying hypergraph independent sets. Specifically, the Glauber dynamics is known to mix rapidly on independent sets in a $k$-uniform hypergraph of maximum degree $Delta$ provided that $Delta lesssim 2^{k/2}$. On the other hand, the best-known zero-freeness around the point $1$ of the independence polynomial on $k$-uniform hypergraphs requires $Delta le 5$, the same bound as on a graph. By introducing a complex extension of Markov chains, we lift an existing percolation argument to the complex plane, and show that if $Delta lesssim 2^{k/2}$, the Markov chain converges in a complex neighborhood, and the independence polynomial itself does not vanish in the same neighborhood. In the same regime, our result also implies central limit theorems for the size of a uniformly random independent set, and deterministic approximation algorithms for the number of hypergraph independent sets of size $k le alpha n$ for some constant $alpha$.

Problem

• Complex Zeros
• Markov Chain Mixing
• k-uniform Hypergraph Polynomials

Innovation

• Complex Markov Chains
• Partition Function Zeros
• k-uniform Hypergraph Independence Polynomials