Three views on the thinned Bernoulli field on the line

TL;DR

This paper analyzes the behavior of the thinned Bernoulli field on the line, revealing its quasilocality properties, ergodic characteristics, and a new Markov chain construction as the occupation density approaches one.

Contribution

It provides a detailed one-dimensional analysis of the TBF, including asymptotics of boundary sensitivity, ergodic properties, and a novel Markov chain representation.

Findings

TBF on the line is always quasilocally Gibbs.

Sensitivity to boundary conditions increases with p, indicating loss of quasilocality.

The TBF can be represented by a generalized house of cards Markov chain.

Abstract

This paper investigates the thinned Bernoulli field (TBF) on the one-dimensional integer lattice, where isolated occupied sites are removed from a standard Bernoulli configuration with density $p$. Our present work complements previous findings in higher dimensions and on trees by focusing on the detailed behavior on the line, particularly as $p$ approaches $1.$ First we show that while the TBF on the line is always quasilocally Gibbs, it displays a growing sensitivity to boundary conditions as $p$ increases, indicating an incipient loss of quasilocality. We provide precise asymptotics for this phenomenon, which is an echo of non-quasilocality happening in higher dimensions. Second, we turn to the one-sided point of view and prove that the TBF is a g-measure in the sense of dynamical systems and ergodic theory. The corresponding g-function is quasilocal but becomes long-range again for large $p$. From that we finally develop our third view, in which we provide a transparent construction of the process in terms of a driving Markov chain on the integers of generalized house of cards type, offering a novel perspective on the TBF.