The Hard-Core Model on Bipartite Spectral Expanders: Counting and Sampling at All Fugacities

2026-08-04

AI Summary

This work addresses the problem of efficient approximate counting and sampling for the hard-core model at any fugacity λ > 0 on Δ-regular bipartite graphs satisfying a spectral expansion condition. By introducing a quadratically tilted measure that accounts for left-right occupancy imbalance, the authors combine Glauber dynamics, discrete Gaussian identities, truncation, and simulated annealing to handle the low-fugacity regime, while refining polymer model techniques for high fugacity. Phase dominance and convergence of cluster expansions are guaranteed solely in terms of an upper bound σ₂(M_G) on the second-largest singular value of the graph’s adjacency matrix. When σ₂(M_G) ≤ c(Δ² / log(eΔ))^{1/3} for an absolute constant c, the approach yields an FPRAS and an efficient approximate sampler for all λ > 0, covering random Δ-regular bipartite graphs for all sufficiently large Δ and providing instance-specific certificates of success.

Abstract

We study approximate counting and sampling algorithms for the hard-core model on $Δ$-regular bipartite graphs under a spectral expansion condition. Let $M_G$ be the biadjacency matrix of $G$. For every fixed $ξ\in(0,1)$, we give an FPRAS for the hard-core partition function and an efficient approximate sampler whenever \[ λ\leq \frac{1-ξ}{σ_2(M_G)}. \] The main idea is to introduce a family of quadratic tilts in the left-right occupation imbalance and show that each tilted measure can be sampled efficiently using Glauber dynamics. A discrete Gaussian identity expresses the original hard-core model as an exact positive mixture of these tilted measures; truncation and simulated annealing then yield efficient counting and sampling algorithms. For the complementary high-fugacity regime, we refine the polymer-model approach and show that the required phase-dominance and cluster expansion conditions follow from the singular-spectrum bound alone. Combining the two regimes, we obtain efficient approximate counting and sampling at every fugacity $λ>0$ whenever \[ σ_2(M_G)\leq c\left(\frac{Δ^2}{\log(\mathrm eΔ)}\right)^{1/3} \] for an absolute constant $c>0$. In particular, this recovers all-fugacity algorithms for random $Δ$-regular bipartite graphs for all sufficiently large $Δ$, while providing an efficiently verifiable certificate of their success on a given instance.

Problem

• hard-core model
• approximate counting
• approximate sampling
• spectral expanders
• fugacity

Innovation

• hard-core model
• spectral expanders
• approximate counting
• Glauber dynamics
• polymer model