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

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dc.creatorde la Rue, Thierry
dc.creatorFernandez, Roberto
dc.creatorSokal, Alan D.
dc.date2007-04-24
dc.date.accessioned2026-07-07T09:52:27Z
dc.date.available2026-07-07T09:52:27Z
dc.descriptionMotivated 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.
dc.descriptionLaTex2e, 80 pages including 4 figures
dc.identifierhttps://arxiv.org/abs/0704.3156
dc.identifierhttp://arxiv.org/abs/0704.3156
dc.identifierMarkov Processes and Related Fields 14, 1--78 (2008)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/165602
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject60J99 (Primary); 15A48, 31C20, 31C99, 60J10, 60J45, 82B20 (Secondary)
dc.titleHow to clean a dirty floor: Probabilistic potential theory and the Dobrushin uniqueness theorem
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