Operads and the Markov Property on the square lattice

Abstract: Markov processes on the lattices with arbitrary dimension are omnipresent in statistical mechanics; however their algebraic description is complete only in dimension 1, for which linear algebra provides many tools complementary to the probabilistic approach: as an example, invariant measures are eigenvectors of the generator. In larger dimension, such algebraic tools are absent due to the more involved structure of boundaries. The present work fills this gap by providing a new and complete algebraic description of these models without any other assumption than the Markov property. In order to handle higher dimensions and higher products, the language of operads is used in order to focus on associativities and the geometric interpretation of the algebraic products. This formalism leads to a new parametrization of boundary conditions of Markov processes, with concrete computations. This parametrization is inspired matrix product states in the physics literature. Among others, a probabilistic application elaborated in this paper is the construction of translation-invariant infinite-volume Gibbs measures on the whole lattice by using Kolmogorov's extension; this provides a new alternative tool to the traditional analytical approaches of large size limits. Various models are considered as illustrations.