谱条件下伊辛模型的自由能
Free energy of Ising models under a spectral condition
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中文总结 AI 辅助
该研究证明了加权图的伊辛模型自由能密度在满足谱条件时为左收敛拓扑下的局部函数，并推导了自旋玻璃等模型在局部树状图上自由能密度的新极限定理。
中文摘要 AI 辅助
以顶点数N为索引的稀疏加权图序列$(G_N:N\ge1)$，若所有（经适当加权的）子图计数在$N\to\infty$时收敛到极限，则称其为左收敛的，该概念自然推广了Benjamini-Schramm的局部弱收敛定义。广泛的研究议程旨在确定哪些“全局”图性质由左收敛或局部弱收敛决定（即这些性质中哪些实际上是局部的）。我们证明，在图$G_N$的加权邻接矩阵${\boldsymbol A}_N$满足谱条件时，该加权图上伊辛模型的自由能密度在左收敛拓扑中是连续的（因此是局部函数）。我们的证明将伊辛测度分解为乘积测度的对数凹组合，并利用对数凹测度的朗之万动力学快速混合性质。作为应用，我们推导了自旋玻璃、反铁磁体及磁化受限铁磁模型在局部树状图上自由能密度的新极限定理。
英文摘要
A sequence of sparse weighted graphs $(G_N:N\ge 1)$ indexed by the number of vertices $N$ is said to be left-convergent if all (suitably weighted) subgraph counts converge to a limit as $N\to\infty$. This notion generalizes in a natural way Benjamini-Schramm's definition of local weak convergence. A broad research agenda aims at determining which `global' graph properties are determined by left or local weak convergence (in other words, which of these properties are in fact local). We prove that, under a spectral condition on the weighted adjacency matrix ${\boldsymbol A}_N$ of graph $G_N$, the free energy density of the Ising model on this weighted graph is continuous in the left convergence topology (and hence is a local function). Our proof uses the decomposition of the Ising measure as a log-concave combination of product measures, and of the rapid mixing of Langevin dynamics for log-concave measures. As applications, we derive new limit theorems for the free energy density of spin glasses, antiferromagnets, and magnetization constrained ferromagnetic models, on locally tree-like graphs.
