The large deviation principle (LDP) is studied 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, and sufficient conditions for uniqueness of the optimizer and existence of constant optimizers are given.
Abstract
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, Erd\H{o}s-R\'enyi 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.
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