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

TL;DR

This paper explores probabilistic potential theory related to the Dobrushin uniqueness theorem, analyzing conditions for the convergence of cleaning operators and their relation to dirt distribution in statistical mechanics models.

Contribution

It introduces a novel matrix and tree formalism to analyze the convergence of cleaning operators and extends the Dobrushin theorem to new settings with infinite index sets.

Findings

Convergence of cleaning operators under infinite sum conditions

Introduction of a tree formalism for potential theory analysis

Characterization of dirt distribution after cleaning processes

Abstract

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.