2025/12/09 by Kuelske, Christof, Schubert, Niklas
Mathematics · Physics and Astronomy · #60K35 #82B26 #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.2512.08800
openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28
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.