无序单体-二聚体模型中的中心极限定理
Central Limit Theorem in Disordered Monomer-Dimer Model
AI总结：
该研究将无序单体-二聚体模型的中心极限定理从柱图推广到有界度一般有限图，在不同矩条件与体积增长假设下证明自由能及二聚体数吉布斯平均的高斯CLT并给出收敛速率，基于高概率关联衰减思路完成证明。
摘要：
我们研究具有有界度的一般有限图上的无序单体-二聚体模型。在权重分布满足有限四阶矩的假设下，我们证明了相关吉布斯测度的自由能满足高斯中心极限定理，并给出了收敛速率。若进一步假设底层图具有一致次指数体积增长，那么在权重分布满足近乎最优的有限$(2+ε)$阶矩假设下，该中心极限定理仍然成立。这一结果推广了Dey和Krishnan（arXiv:2109.12716）的最新结论，他们证明了柱图上无序单体-二聚体模型的高斯中心极限定理。我们的证明基于如下思路：无序单体-二聚体模型以高概率呈现关联衰减特性。此外，我们还建立了底层图具有次指数体积增长且边权重为高斯分布时，二聚体数的吉布斯平均的中心极限定理。
英文摘要：
We consider the disordered monomer-dimer model on general finite graphs with bounded degrees. Under the finite fourth moment assumption on the weight distributions, we prove a Gaussian central limit theorem for the free energy of the associated Gibbs measure with a rate of convergence. The central limit theorem continues to hold under a nearly optimal finite $(2+ε)$-moment assumption on the weight distributions if the underlying graphs are further assumed to have a uniformly subexponential volume growth. This generalizes a recent result by Dey and Krishnan (arXiv:2109.12716) who showed a Gaussian central limit theorem in the disordered monomer-dimer model on cylinder graphs. Our proof relies on the idea that the disordered monomer-dimer model exhibits a decay of correlation with high probability. We also establish a central limit theorem for the Gibbs average of the number of dimers where the underlying graph has subexponential volume growth and the edge weights are Gaussians.
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