Counting and Sampling Anti-Ferromagnetic Potts Models on Random Regular Bipartite Graphs in the Non-uniqueness Regime

Abstract: The anti-ferromagnetic multi-state Potts model, a generalization of the Ising model, is one of the most fundamental models in statistical physics. It was conjectured by Kotecký (Phys.~Rev.~B, 1985) that the model undergoes a phase transition from a disordered phase at infinite temperature to an ordered phase at sufficiently low temperature on lattices. Such phase transitions are believed to play an important role in computational complexity theory and remain closely connected to the problem of approximating the partition function of the system. For proper three-coloring models (corresponding to the zero-temperature), torpid mixing of a family of local-update Markov chains on lattices was established by Galvin, Kahn, Randall and Sorkin (SIDMA, 2015), coinciding with the presence of phase coexistence following shown by Feldheim and Spinka (J.~Eur.~Math.~Soc., 2019). In this work, we study approximating the partition function of the anti-ferromagnetic multi-state Potts model at low temperature on random regular bipartite graphs, which are with high probability good bipartite expanders. On the negative side, we generalize the result by Geisler, Kang, Sarantis and Wdowinski (arXiv, 2026) for anti-ferromagnetic Ising models to show that when the temperature is sufficiently low relative to the degree of the underlying graph, the celebrated single-site Glauber dynamics has exponentially slow mixing time. On the positive side, we design a deterministic algorithm that yields an approximation to the partition function of the model via the framework of abstract polymer models as Jenssen, Keevash and Perkins (SICOMP, 2020), Liao, Lin, Lu and Mao (Theor.~Comput.~Sci., 2022), Galanis, Goldberg and Stewart (TOCT, 2021) and Geisler, Kang, Sarantis and Wdowinski (arXiv, 2026).

• The paper shows that single-site Glauber dynamics exhibit exponential slow mixing for anti-ferromagnetic Potts models on bipartite graphs in the non-uniqueness regime.
• It introduces a deterministic FPTAS based on abstract polymer models and cluster expansion techniques to efficiently approximate the model’s partition function.
• The study links the ordered phase structure to algorithmic tractability, highlighting how phase transitions influence computational hardness in complex spin systems.

Summary of "Counting and Sampling Anti-Ferromagnetic Potts Models on Random Regular Bipartite Graphs in the Non-uniqueness Regime" (2606.21250)

Anti-Ferromagnetic Potts Models and Computational Complexity

The $q$-state anti-ferromagnetic Potts model on graphs generalizes the Ising model, capturing a range of statistical physics phenomena, particularly phase transitions between disordered and ordered states depending on the inverse temperature parameter $\beta$. When implemented on random regular bipartite graphs—bipartite expanders with high probability—these models are crucial in studying connections between complex combinatorics, phase transition phenomena, and computational hardness.

Exact computation of the Potts partition function $Z_{G,q}(\beta)$ is $\#P$-complete even for sparse, regular triangle-free graphs. Thus, the theoretical investigation focuses on the existence and construction of efficient approximation schemes (FPTAS/FPRAS) for this partition function, contingent on the underlying phase structure of the Gibbs distribution.

Main Results: Slow Mixing of Glauber Dynamics and Deterministic Approximation

Slow Mixing of Glauber Dynamics

The paper rigorously proves that single-site Glauber dynamics—canonical local-update Markov chains—exhibit exponentially slow mixing for the anti-ferromagnetic $q$-Potts model on random regular bipartite graphs at low temperatures (large $\beta$ and large $\Delta$). There exist explicit constants $C_1, C_2 > 0$ such that for $q, \Delta \ge 3$, $\Delta > C_1 q\ln{q}$, and $\beta$0, the mixing time is at least $\beta$1 for some $\beta$2 as $\beta$3 (2606.21250). This torpid mixing is attributed to phase coexistence and the geometric concentration of configurations around ordered coloring patterns, analogous to phenomena established for $\beta$4-coloring models on lattices [GKRS15, FS19].

Deterministic FPTAS via Abstract Polymer Models

Despite slow mixing, the authors present a deterministic FPTAS for the Potts partition function in this non-uniqueness regime. Utilizing ordered phase structure and combinatorial expansion properties, the algorithm leverages the abstract polymer model with cluster expansion techniques, validating the Kotecký-Preiss condition and achieving fully polynomial approximation in $\beta$5 time. The result holds under conditions: $\beta$6, $\beta$7, and $\beta$8.

Technical Contributions

Ordered Phases and Configuration Geometry

The configuration space of the anti-ferromagnetic $\beta$9-Potts model, at low temperatures, exhibits concentration near "ordered patterns," namely partitions $Z_{G,q}(\beta)$0 of color sets where most vertices in a bipartition are assigned disjoint subsets. This phase structure, formalized via configuration ranks and pattern definitions, underlies both the conductance-based slow mixing argument and the decomposition required for efficient partition function approximation. The geometric concentration is evidenced by exponential tail bounds on the measure of "high-rank" or unstructured configurations.

Conductance Analysis and Slow Mixing

Through rigorous conductance arguments, the authors construct low-conductance subsets in the configuration space, exploiting the expansion properties of random regular bipartite graphs. The Cheeger-type lower bounds on mixing time, in conjunction with detailed combinatorial and measure estimates, demonstrate exponentially slow mixing for Glauber dynamics. Extending prior Ising model results [GKSW26], this analysis overcomes hurdles in bounding weights of "balanced" configurations for general $Z_{G,q}(\beta)$1-Potts systems.

Polymer Models for FPTAS

For deterministic approximate counting, the work builds a polymer model indexed by sparse, low-rank subsets and patterns, validating the required convergence via combinatorial expansion and weight bounds. The algorithmic framework combines enumeration of compatible polymers (connected subsets in the $Z_{G,q}(\beta)$2 power graph) and multiplication rules for configuration weights. The verification of the Kotecký-Preiss condition ensures cluster expansion convergence and enables efficient deterministic approximation even far in the non-uniqueness regime. Comparisons with related work show improved degree/color constraints and adaptation to the anti-ferromagnetic context.

Implications, Future Directions, and Open Problems

This paper establishes a sharp dichotomy: slow local mixing of Glauber-type chains versus existence of FPTAS for partition function approximation in non-uniqueness, ordered phase regimes for anti-ferromagnetic Potts models on random regular bipartite graphs. The separation between phase coexistence and algorithmic hardness—in contrast to lattice models and non-bipartite expanders—deepens current understanding of the computational-combinatorial landscape for spin systems.

Key theoretical implications include:

• Phase structure governs algorithmic tractability: Ordered phases facilitate deterministic approximation algorithms despite the absence of rapid mixing.
• Conductance-based analysis is central for slow mixing proofs: Expansion properties and geometric concentration provide generic tools for analyzing Markov chains in complex spin systems.
• Abstract polymer methodologies generalize efficiently to bipartite expanders: Adaptation of cluster expansion and polymer enumeration extends FPTAS landscapes for models beyond proper colorings and hard-core systems.

Practically, these results inform efficient simulation and approximate inference for large-scale statistical physics models on expander topologies, relevant for applications in statistical mechanics, randomized algorithms, and combinatorial optimization.

Future research directions:

• Phase transition characterization: Determining precise boundaries and thresholds for ordered phase emergence in random regular bipartite graphs for general $Z_{G,q}(\beta)$3.
• Rapid Markov chain design: Investigating polymer dynamics or global-update chains that could achieve rapid mixing in the ordered phase regime.
• Algorithmic extension to other random bipartite models: Establishing FPTAS for partition function approximation in Erdős–Rényi bipartite graphs or unbounded degree settings, beyond controlled expansion.

Conclusion

The paper provides a comprehensive analysis of the interplay between phase structure, Markov chain mixing, and deterministic approximation algorithms for the anti-ferromagnetic $Z_{G,q}(\beta)$4-Potts model on random regular bipartite graphs. Through geometric, combinatorial, and algorithmic techniques, it demonstrates both torpid local mixing and existence of efficient deterministic approximation under ordered phase regimes. This advances both theoretical understanding and algorithmic approaches for counting in complex spin systems.