发表机构
Stanford University; National University of Singapore（斯坦福大学; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非均匀随机图上Ising模型对数配分函数的高阶波动，证明在无自环高温区域满足N尺度CLT，方差含所有阶循环贡献，与自环情形不同。
AI 中文摘要
经典中心极限定理（CLT）适用于Erdős-Rényi随机图上Ising模型的对数配分函数，但在无自环的高温区域会导致退化极限。本文在更一般的非均匀随机图上的Ising模型背景下填补了这一令人惊讶的空白。特别地，我们证明：若$G(N,W)$是由图限$W$生成的无自环非均匀随机图，则在高温区域，相应Ising模型的对数配分函数满足$N$尺度上的中心极限定理，其极限方差包含所有阶循环的贡献。这一行为与允许自环的相应模型形成鲜明对比，后者在$\sqrt{N}$尺度上发生类似的高斯波动，其渐近方差完全由$W$的对角轮廓决定。作为独立有趣的副产品，我们的分析还得到了相关预解矩阵的对数行列式的中心极限定理，该定理通过结合三角阵列的Lindeberg中心极限定理与高阶迹项的适当界来建立。
英文摘要
Classical central limit theorems (CLTs) for log-partition functions of Ising models on Erdős-Rényi random graphs lead to degenerate limits in the high-temperature regime when self-loops are absent. In this paper, we fill this surprising gap in the more general context of Ising models on inhomogeneous random graphs. In particular, we show that if $G(N,W)$ is an inhomogeneous random graph without self-loops generated by a graphon $W$, then, in the high-temperature regime, the log-partition function of the corresponding Ising model satisfies a CLT on the $N$-scale, with a limiting variance that incorporates contributions from cycles of all orders. This behavior contrasts sharply with that of the corresponding model in which self-loops are allowed, where the analogous Gaussian fluctuations occur on the $\sqrt{N}$-scale, with their asymptotic variance determined entirely by the diagonal profile of $W$. Our analysis yields, as a byproduct of independent interest, a CLT for the log-determinant of the associated resolvent matrix, established by combining the Lindeberg CLT for triangular arrays with suitable bounds on higher-order trace terms.
Comments39 pages
