Theory of computation of multidimensional entropy with an application to the monomer-dimer problem

Consider all colorings of a finite box in a multidimensional grid with a given number of colors subject to given local constraints. We outline the most recent theory for the computation of the exponential growth rate of the number of such colorings as a function of the dimensions of the box. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 9 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%.