Decompositions, Traces and Excursions

We study the decompositions induced by splitting the set X of graph vertices in two parts, D and F, the energy and the Markov chain being decomposed into its restriction to D and its trace on F. It is often the case that F can be viewed as a boundary of D, and then we are addressing a problem of boundary decomposition. Paths are decomposed into their trace on F and a set of excursions into D. Excursion decomposition is applied to loop ensembles. The case in which F is reduced to a single point receives special attention.