Three views on the thinned Bernoulli field on the line

TL;DR

The paper analyzes the thinned Bernoulli field on the line, obtained by removing isolated occupied sites from a Bernoulli configuration with density $p$, and reveals three coherent perspectives: a two-sided Gibbs (quasilocal) description with an explicit image specification, a one-sided g-measure description with an explicit g-function, and a driving generalized House of Cards (GHoC) Markov-chain representation. It constructs an explicit two-sided specification $\gamma'_p$, proves exponential decay of boundary influence in terms of the eigenvalue ratio $a(p)$, and shows uniqueness via a finite-energy argument; it then derives the g-function $g_p(n)$ for the one-sided view, proves uniqueness of the corresponding g-measure, and connects this with a GHoC process having a unique stationary distribution and yielding a deterministic map to the TBF. Parity effects arise in the large-$p$ limit, producing echoes of non-quasilocality on the line, while the GHoC representation provides a fast route to understanding uniqueness and long-range structure. Together, these results illuminate how a simple local map can generate nontrivial long-range features in one dimension and offer a unified framework for extensions to higher dimensions and related transformed-point processes.

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.

Figures (13)

• Figure 1: An example of a spin configuration on the line and the application of the map $T$. Every coloured dot is an occupied site, while every uncoloured dot is unoccupied. The blue coloured dots mark the isolated occupied sites.
• Figure 2: A second-layer configuration $\omega' \in \Omega'$ on the line $\mathbb{Z}$. Clusters of occupied sites $\Theta$ prescribed by $\omega'$ are in full orange, all other sites are empty in $\omega'$. The fixed area $\overline{\Theta}$ of this configuration is highlighted in orange and the remaining half-black and half-white colored dots correspond to the unfixed areas in $\mathscr{U}$. They may be occupied or unoccupied, while obeying the isolation constraint, in the underlying first-layer configuration.
• Figure 3: Depicted are plots the eigenvalue ratio $a=a(p)$ and the $g$-function $g_p$ as a function of the occupation density $p$ describing the behavior of the TBF.
• Figure 4: A non-isolated configuration $\omega'$ on the line, highlighted in blue, together with its corresponding GHoC path $(n_i)_{i\in \mathbb{Z}}$ in $\mathbb{N}_0\cup \{\infty\}$, where $n_i$ is indicated below each site $i$. For each site $i\in \mathbb{Z}$, the state shown below indicates the distance $n_i$ to the next pair of occupied sites in its past. Note that, if the configuration in the past ends with the pair "empty followed by occupied", the distance to the next double one is put to be $0$, as the example shows in the middle of the triplet of occupied sites.
• Figure 5: Depicted is the GHoC Markov process with state space $\mathbb{N}_0\cup \{\infty\}$ for the first 7 states. The transition probabilities are given by the g-function $g_p$ of the TBF. Furthermore, the map $\tau$ is depicted in blue which provides a one-to-one correspondence between a non-isolated configuration $\omega'$ and its GHoC path $(n_i)_{i\in \mathbb{Z}}$ taking values in $\mathbb{N}_0\cup\{\infty\}$. Note that an example of this correspondence is given in Figure \ref{['fig: D2D1 sequence']}.

Theorems & Definitions (30)

• Remark 1
• Definition 1
• Theorem 1
• Remark 2
• Theorem 2
• Remark 3
• Proposition 1
• Theorem 3
• Lemma 1
• Proposition 2