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.

41 pages