How to Clean a Dirty Floor: Probabilistic Potential Theory and the Dobrushin Uniqueness Theorem

Motivated by the Dobrushin uniqueness theorem in statistical mechanics, we consider the following situation: Let α be a nonnegative matrix over a finite or countably infinite index set X, and define the “cleaning operators” βh = I1−h + Ihα for h: X → [0, 1] (here If denotes the diagonal matrix with entries f). We ask: For which “cleaning sequences” h1, h2, . . . do we have cβh1 · · · βhn → 0 for a suitable class of “dirt vectors” c? We show, under a modest condition on α, that this occurs whenever ∑ i hi = ∞ everywhere on X. More generally, we analyze the cleaning of subsets Λ ⊆ X and the final distribution of dirt on the complement of Λ. We show that when supp(hi) ⊆ Λ with ∑ i hi = ∞ everywhere on Λ, the operators βh1 · · · βhn converge as n → ∞ to the “balayage operator” ΠΛ = ∑∞ k=0(IΛα) IΛc . 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 α are treated as noncommuting indeterminates.