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Phase Transitions via Complex Extensions of Markov Chains

Jingcheng Liu, Chunyang Wang, Yitong Yin, Yixiao Yu

TL;DR

The paper investigates phase transitions by locating complex zeros of partition functions through complex extensions of Markov chains. It develops a complex Glauber-dynamics framework and a percolation-based analysis to prove zero-free regions for the hypergraph independence polynomial, achieving regimes matching the algorithmic mixing threshold $\Delta \lesssim 2^{k/2}$ (up to poly$(k)$ factors). It further derives a central limit theorem for the size of a random independent set and provides a deterministic approximation scheme for counting independent sets of a given size, along with a Fisher-to-Lee–Yang zero reduction. Collectively, these results bridge probabilistic mixing, algebraic properties of partition functions, and deterministic counting, yielding new tools for understanding phase transitions in hypergraph models and beyond.

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 $Δ$ provided that $Δ\lesssim 2^{k/2}$. On the other hand, the best-known zero-freeness around the point $1$ of the independence polynomial on $k$-uniform hypergraphs requires $Δ\le 5$, 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 $Δ\lesssim 2^{k/2}$, 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 $k \le αn$ for some constant $α$.

Phase Transitions via Complex Extensions of Markov Chains

TL;DR

The paper investigates phase transitions by locating complex zeros of partition functions through complex extensions of Markov chains. It develops a complex Glauber-dynamics framework and a percolation-based analysis to prove zero-free regions for the hypergraph independence polynomial, achieving regimes matching the algorithmic mixing threshold (up to poly factors). It further derives a central limit theorem for the size of a random independent set and provides a deterministic approximation scheme for counting independent sets of a given size, along with a Fisher-to-Lee–Yang zero reduction. Collectively, these results bridge probabilistic mixing, algebraic properties of partition functions, and deterministic counting, yielding new tools for understanding phase transitions in hypergraph models and beyond.

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 -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 -uniform hypergraph of maximum degree provided that . On the other hand, the best-known zero-freeness around the point of the independence polynomial on -uniform hypergraphs requires , 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 , 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 for some constant .

Paper Structure

This paper contains 33 sections, 32 theorems, 137 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Hypergraph independence polynomial
  3. Our contributions
  4. Technical overview
  5. Locating complex zeros through marginal measures
  6. Marginal measures through information percolation on complex Markov chains
  7. Convergence of the complex systematic scan Glauber dynamics
  8. Preliminaries and notations
  9. Complex normalized measures
  10. Graphical model and complex zeros
  11. Glauber dynamics: random scan and systematic scan
  12. $2$-tree
  13. Convergence of complex Markov chains
  14. Complex extensions of Markov chains
  15. Complex Markov chain
  16. ...and 18 more sections

Key Result

Theorem 1.1

Fix $k\ge 2$ and $\Delta\ge 3$. Let $0<\varepsilon<\frac{1}{9k^5\Delta^2}$, and let $[0,\lambda_{c,\varepsilon})$ be the real segment such that the following holds for all $\lambda\in [0,\lambda_{c,\varepsilon})$: Let $\mathcal{D}_{\varepsilon}$ be the union of $\varepsilon$-balls around the segment given by $\mathcal{D}_{\varepsilon}=\left\{z\in\mathbb{C}\mid \exists \lambda\in[0,\lambda_{c,\var

Figures (1)

  • Figure 1: A rough summary of connections between three types of phase transitions. We do not distinguish the exact form of phase transitions within each type.

Theorems & Definitions (69)

  • Theorem 1.1: Lee-Yang zeros of hypergraph independence polynomial
  • Corollary 1.2
  • Theorem 1.3: Fisher zeros of hypergraph independence polynomial
  • Remark 1.4: implications for deterministic counting
  • Theorem 1.5: Central limit theorem for hypergraph independent sets
  • Theorem 1.6
  • Theorem 1.7: Convergence of systematic scan Glauber dynamics with complex transitions
  • Definition 2.1: $2$-tree
  • Definition 2.2: construction of a maximal $2$-tree in a connected graph Vishesh21towards
  • Lemma 2.3: borgs2013left,feng2021rapid
  • ...and 59 more