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

de la Rue, Thierry; Fernandez, Roberto; Sokal, Alan D.

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

Authors

de la Rue, Thierry

Fernandez, Roberto

Sokal, Alan D.

Description

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.
LaTex2e, 80 pages including 4 figures