A Combinatorial Analysis of Interacting Diffusions

We attempt to unify the analysis of several families of naturally occurring multidimensional stochastic processes by studying the underlying combinatorics involved. At equilibrium, the behavior of these processes is determined by the properties of a randomly chosen point of a corresponding polyhedron. How such a randomly chosen point behaves is a difficult question which is intertwined with the geometry and the symmetry of the polyhedron. The simplest of all cases is the simplex where a complete probabilistic study is known. A possible general strategy is through triangulation of the polyhedron where we decompose it as a union of simplices with non-intersecting interiors. In particular we study the case when the polyhedron is a simplicial polytope, since they correspond to the natural examples of stochastic processes. This is the case when the polytope is invariant under a Coxeter group action, which leads to a simple and explicit description of the equilibrium behavior of the stochastic processes in terms of independent and identically distributed Exponential random variables. Another class of examples is furnished by processes indexed by weighted graphs, all of which generate simplicial polytopes with n! faces. We show that the proportion of volume contained in each component simplex corresponds to a probability distribution on the group of permutations, some of which have surprising connections with the classical urn models.