二分谱扩张图上的硬核模型：在所有逸度下的计数与采样
The Hard-Core Model on Bipartite Spectral Expanders: Counting and Sampling at All Fugacities
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中文总结 AI 辅助
该研究针对满足谱扩张条件的Δ-正则二分图上的硬核模型，结合二次倾斜测度与聚合物模型方法，实现了所有逸度下的高效近似计数与采样算法。
中文摘要 AI 辅助
我们研究Δ-正则二分图上硬核模型的近似计数与采样算法，该图满足谱扩张条件。设M_G为图G的双邻接矩阵，对于每个固定的ξ∈(0,1)，当λ≤(1−ξ)/σ₂(M_G)时，我们给出硬核配分函数的完全多项式随机近似方案（FPRAS）及高效近似采样器。核心思路是引入左右占据失衡的二次倾斜族，证明每个倾斜测度均可通过Glauber动力学高效采样；离散高斯恒等式将原始硬核模型表示为这些倾斜测度的精确正混合，再结合截断与模拟退火，得到高效计数与采样算法。对于互补的高逸度 regime，我们改进聚合物模型方法，证明所需的相位主导性与簇展开条件仅由奇异谱界即可推出。结合两种 regime，当σ₂(M_G)≤c(Δ²/(eΔ))^(1/3)（c为绝对常数且c>0）时，可对所有逸度λ>0实现高效近似计数与采样。特别地，这为所有足够大Δ的随机Δ-正则二分图恢复了全逸度算法，同时为给定实例提供了可高效验证的成功证书。
英文摘要
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.
