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Article Dans Une Revue Markov Processes And Related Fields Année : 2008

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

Résumé

Motivated by the Dobrushin uniqueness theorem in statistical mechanics, we consider the following situation: Let \alpha be a nonnegative matrix over a finite or countably infinite index set X, and define the "cleaning operators" \beta_h = I_{1-h} + I_h \alpha for 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 \beta_{h_1} ... \beta_{h_n} \to 0 for a suitable class of "dirt vectors" c? We show, under a modest condition on \alpha, that this occurs whenever \sum_i h_i = \infty everywhere on X. More generally, we analyze the cleaning of subsets \Lambda \subseteq X and the final distribution of dirt on the complement of \Lambda. We show that when supp(h_i) \subseteq \Lambda with \sum_i h_i = \infty everywhere on \Lambda, the operators \beta_{h_1} ... \beta_{h_n} converge as n \to \infty to the "balayage operator" \Pi_\Lambda = \sum_{k=0}^\infty (I_\Lambda \alpha)^k I_{\Lambda^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 \alpha are treated as noncommuting indeterminates.

Dates et versions

hal-00267997 , version 1 (30-03-2008)

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Citer

Thierry de La Rue, Roberto Fernandez, Alan D. Sokal. How to clean a dirty floor: Probabilistic potential theory and the Dobrushin uniqueness theorem. Markov Processes And Related Fields, 2008, 14, pp.1-78. ⟨hal-00267997⟩
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