Models with exclusions: Discretization and perfect simulation

Models in continuum space are usually simulated by discretizing the underlying space. The question arises as to how trustable is such a procedure from a rigorous mathematical viewpoint. Is the discretized distribution close to the continuum one if the discretization step is small enough? More generally, is the phase diagram of the discretized system similar to that of the original continuum system? For systems defined by hard exclusions -e.g. Widom Rowlison models- we are able to prove closeness between the continuous and discretized statistical-mechanical descriptions in regimes without phase transition. More precisely, the models must satisfy a condition allowing the convergence of the so-called "ancestor's algorithm". This algorithm -which can be interpreted as a probabilistic version of cluster expansions -yields, in addition, a perfect simulation scheme coupling the continuum model with its discretizations. Tomasz Schreiber understood this algorithm better than its authors and exploited it to develop a sophisticated theory of polygonal Markov fields. He would have been a natural coauthor of our work.

Tomasz Schreiber's Memorial Session