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 through a functional fixed-point equation. As applications, we consider monochromatic subgraph counts in sparse random graphs, Erdős–Rényi hypergraphs, and a generalized Potts statistic of order at least two. We also give sufficient conditions for uniqueness of the optimizer and for replica symmetry and obtain universal weak laws for a broad class of tensor Gibbs models with approximately regular tensors.
This is joint work with Sohom Bhattacharya, Nabarun Deb, and Sumit Mukherjee.