Decay of correlations for the hardcore model on the d-regular random graph
Decay of correlations for the hardcore model on the d-regular random graph
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DOI：
10.1214/16-ejp3552
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发表时间：
2016-01-01
影响因子：
1.4
通讯作者：
Tetali, Prasad
中科院分区：
文献类型：
--
作者：
Bhatnagar, Nayantara;Sly, Allan;Tetali, Prasad
统计物理学中关于随机图上自旋系统的一个关键洞见是吉布斯树测度所起的核心作用。我们确定了硬核模型在随机正则图上的局部弱极限，直至最大独立集的密度，该密度由凝结阈值所包围并渐近地接近。我们证明了核测度在强意义上局部概率收敛于树上的自由边界条件Gibbs测度。因此，我们证明了随机图上的重建阈值，指示点开始设置空间相关性，等于我们确定精确渐近的d规则树上的重建阈值。我们期望我们的方法可以推广到第二矩方法适用的大范围自旋系统。
A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on random regular graphs upto a density for the largest independent set that is bounded by and goes asymptotically to the condensation threshold. We show that the hardcore measure converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence we show that the reconstruction threshold on the random graph, indicative of the onset of point to set spatial correlations, is equal to the reconstruction threshold on the d-regular tree for which we determine precise asymptotics. We expect that our methods will generalize to a wide range of spin systems for which the second moment method holds.