English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Motivated by the Dobrushin uniqueness theorem in statistical mechanics, weconsider the following situation: Let \alpha be a nonnegative matrix over afinite 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 thediagonal 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 occurswhenever \sum_i h_i = \infty everywhere on X. More generally, we analyze thecleaning of subsets \Lambda \subseteq X and the final distribution of dirt onthe 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 areobtained in two ways: by a fairly simple matrix formalism, and by a morepowerful tree formalism that corresponds to working with formal power series inwhich the matrix elements of \alpha are treated as noncommuting indeterminates.

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