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 $\Delta\leq 2^{k/2}$. On the other hand, the best-known zero-freeness around the point 1 of the independence polynomial on k-uniform hypergraphs requires $\Delta\leq 5$, 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 $\Delta\leq 2^{k/2}$., 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≤αn for some constant $\alpha$.
Jingcheng Liu
Associate Professor
I am an Associate Professor in the Theory Group of the Department of Computer Science and Technology at Nanjing University. I am broadly interested in theoretical computer science.
Chunyang Wang
Ph.D Student
I am currently a fifth-year Ph.D student in the Theory Group in the Department of Computer Science and Technology at Nanjing University. My research interest lies in a broad aspect of computer science. Currently, I am focusing on algorithms for counting and sampling.
Yitong Yin
Professor
I am a professor in the Theory Group in the Department of Computer Science and Technology at Nanjing University. I am interested in Theoretical Computer Science.