How to clean a dirty floor: Probabilistic potential theory and the Dobrushin uniqueness theorem

Abstract

Motivated by the Dobrushin uniqueness theorem in statistical mechanics, we consider the following situation: Let αbe a nonnegative matrix over a finite or countably infinite index set X, and define the "cleaning operators" β_h = I_{1-h} + I_h αfor h: X \to [0,1] (here I_f denotes the diagonal matrix with entries f). We ask: For which "cleaning sequences" h_1, h_2, ... do we have c β_{h_1} ... β_{h_n} \to 0 for a suitable class of "dirt vectors" c? We show, under a modest condition on α, that this occurs whenever \sum_i h_i = \infty everywhere on X. More generally, we analyze the cleaning of subsets Λ\subseteq X and the final distribution of dirt on the complement of Λ. We show that when supp(h_i) \subseteq Λwith \sum_i h_i = \infty everywhere on Λ, the operators β_{h_1} ... β_{h_n} converge as n \to \infty to the "balayage operator" Π_Λ= \sum_{k=0}^\infty (I_Λα)^k I_{Λ^c). These results are obtained in two ways: by a fairly simple matrix formalism, and by a more powerful tree formalism that corresponds to working with formal power series in which the matrix elements of αare treated as noncommuting indeterminates.