LDP for Inhomogeneous U-Statistics

TL;DR

This paper establishes a Large Deviation Principle for inhomogeneous U/V-statistics, enabling analysis of complex statistical models like tensor Ising and Potts models with new variational characterizations.

Contribution

It introduces a general LDP framework for inhomogeneous U/V-statistics and applies it to Gibbs measures, including tensor generalizations of classical models.

Findings

Derived LDP for inhomogeneous U/V-statistics of general order.

Expressed rate functions as variational problems over function spaces.

Established scaling limits and weak laws for Gibbs measures with tensor Hamiltonians.

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.