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LDP for Inhomogeneous U-Statistics

ANNALS OF APPLIED PROBABILITY(2024)

Univ Florida | Univ Chicago | Columbia Univ

Cited 1|Views8
Abstract
In this paper we derive a Large Deviation Principle (LDP) for inhomogeneous U/V-statistics of a general order. Using this, we derive a LDP for two types of statistics: random multilinear forms, and number of monochromatic copies of a subgraph. We show that the corresponding rate functions in these cases can be expressed as a variational problem over a suitable space of functions. We use the tools developed to study Gibbs measures with the corresponding Hamiltonians, which include tensor generalizations of both Ising (with non-compact base measure) and Potts models. For these Gibbs measures, we establish scaling limits of log normalizing constants, and weak laws in terms of weak* topology, which are of possible independent interest.
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Graph limits,large deviations,U-Statistics,tensor Ising/Potts model
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要点】:本文推导了非均匀U/V统计量的一般阶LDP,并由此得出随机多元线性形式和子图同色副本数量的统计量的LDP,其率函数可以表示为适当函数空间的变分问题。

方法】:利用这些工具研究相应的哈密顿量,包括Ising(具有非紧支测度)和Potts模型的张量推广。

实验】:对这类Gibbs测度,确定了对数规范常数的标度极限和在弱*拓扑结构下的弱定律。