发表机构
Columbia University; University of Florida; University of Chicago Booth School of Business（哥伦比亚大学; 佛罗里达大学; 芝加哥大学布斯商学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究张量加权泛函的大偏差原理，分析张量值吉布斯测度，结合具体例子给出最优解相关条件，证明一类张量吉布斯模型的通用弱定律。
AI 中文摘要
本文研究独立同分布随机变量的张量加权泛函的大偏差原理（LDP），其中张量序列在“坏”割范数的变体下收敛。利用该LDP，我们分析具有张量值哈密顿量的吉布斯测度，并通过泛函不动点方程刻画极限变分问题的最优解。作为应用，我们聚焦几个具体例子，包括稀疏随机图中的单色子图计数、Erdős-Rényi超图，以及阶数$v\ge2$的广义Potts统计量。通过研究该优化问题，我们给出最优解唯一性的充分条件，以及常数最优解（复制对称）的存在条件。结果表明，对于一类具有近似正则张量的张量吉布斯模型，存在通用弱定律。
英文摘要
In this paper, we study the large deviation principle (LDP) for a tensor-weighted functional of i.i.d. random variables, when the sequence of tensors converges under a variant of the "bad" cut norm. Using the LDP, we analyze a Gibbs measure with a tensor-valued Hamiltonian, and characterize the optimizers of the limiting variational problem in terms of a functional fixed point equation. As applications, we focus on several concrete examples, which include monochromatic subgraph counts in sparse random graphs, Erdős-Rényi hypergraphs, and a generalized Potts statistic of order $v\ge 2$. Studying the optimization problem, we give sufficient conditions for uniqueness of the optimizer, as well as for existence of constant optimizers (replica symmetry). Our results demonstrate universal weak laws for a large class of tensor Gibbs models with approximately regular tensors.
