随机图上的硬核模型
The hard-core model on random graphs revisited
摘要：
我们重新审视经典的硬核模型，也被称为独立集和对偶顶点覆盖问题，其中一个粒子的第一邻居硬核排斥的随机图的顶点。虽然分别具有小和非常大的平均度的随机图的情况是相当好的理解，他们产生定性不同的结果，我们在这里的目的是调和这两种情况。我们重新审视了使用（启发式）腔方法可以得到的结果，并表明它为度K>=20的随机正则图上的密度填充的精确密度提供了一个封闭形式的猜想，并且对于K>16，相变的性质与大K相同。这也表明，硬编码模型是结构玻璃和干扰的最简单的平均场晶格模型。
英文摘要：
We revisit the classical hard-core model, also known as independent set and dual to vertex cover problem, where one puts particles with a first-neighbor hard-core repulsion on the vertices of a random graph. Although the case of random graphs with small and very large average degrees respectively are quite well understood, they yield qualitatively different results and our aim here is to reconciliate these two cases. We revisit results that can be obtained using the (heuristic) cavity method and show that it provides a closed-form conjecture for the exact density of the densest packing on random regular graphs with degree K>=20, and that for K>16 the nature of the phase transition is the same as for large K. This also shows that the hard-code model is the simplest mean-field lattice model for structural glasses and jamming.
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