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Markov processes are omnipresent in any dimension, especially in statisticalmechanics; however their algebraic description is complete only in dimension 1(which is often interpreted as "time"), for which linear algebra provides manytools complementary to the probabilistic approach: in particular invariantmeasures are eigenvectors. In larger dimension, such algebraic tools areabsent, in particular because boundaries are more involved. We introduce newoperadic tools to describe Markov processes in any dimension on the squarelattice. The main consequence is a natural description of probabilisticboundary weights in terms of matrix product representations, for which wegeneralize the eigenvector property of 1D invariant measures. This new approachleads to new very interesting algebraic objects, with a local-to-globalperspective. Moreover, finite size Markov processes on the square lattice canalso be seen as a toy case of statistical Euclidean field theory: our algebraictools are related to other categorical approaches to field theory, with theadvantage of providing concrete computational tools.

Operadic approach to Markov processes with boundaries on the square lattice in dimension 2 and larger