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Gibbs Properties of the Bernoulli field on inhomogeneous trees under the removal of isolated sites

Florian Henning, Christof Külske, Niklas Schubert

Abstract

We consider the i.i.d. Bernoulli field $μ_p$ with occupation density $p \in (0,1)$ on a possibly non-regular countably infinite tree with bounded degrees. For large $p$, we show that the quasilocal Gibbs property, i.e. compatibility with a suitable quasilocal specification, is lost under the deterministic transformation which removes all isolated ones and replaces them by zeros, while a quasilocal specification does exist at small $p$. Our results provide an example for an independent field in a spatially non-homogeneous setup which loses the quasilocal Gibbs property under a local deterministic transformation.

Gibbs Properties of the Bernoulli field on inhomogeneous trees under the removal of isolated sites

Abstract

We consider the i.i.d. Bernoulli field with occupation density on a possibly non-regular countably infinite tree with bounded degrees. For large , we show that the quasilocal Gibbs property, i.e. compatibility with a suitable quasilocal specification, is lost under the deterministic transformation which removes all isolated ones and replaces them by zeros, while a quasilocal specification does exist at small . Our results provide an example for an independent field in a spatially non-homogeneous setup which loses the quasilocal Gibbs property under a local deterministic transformation.
Paper Structure (4 sections, 6 theorems, 26 equations, 4 figures)

This paper contains 4 sections, 6 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Part 1 of the proof: Two-layer representation on trees
  4. Part 2 of the proof: Cutsets and pushout method

Key Result

Theorem 1

Consider a countably infinite tree $(V,E)$ which is bounded in the sense that there is a number $2 \leq d_{\text{max}}<\infty$ such that each vertex has at least 3 and at most $d_{\text{max}}+1$ nearest neighbours. Then, there exist $0<p_1(d_{\text{max}})<p_2(d_{\text{max}})<1$ such that for the ind

Figures (4)

  • Figure 1: An example of a spin configuration on the binary tree and the application of the map T. Every coloured dot is an occupied site and every uncoloured dot is unoccupied. The blue coloured dots mark the isolated occupied sites.
  • Figure 2: The configuration $\omega^*\in \Omega$ on the binary tree.
  • Figure 3: An illustration of the type-1 boundary condition $B_5(\rho)$ on the binary tree together with an isolated configuration inside of the ball. The pictured configuration is unoccupied at the root, hence it is adapted to a type-changing cutset $\vec{L} \in \mathscr{C}(0,B_5(\rho))$ of type $0$, which cuts off the rooted subtree, where the configuration resembles the groundstate $\omega^0$, from the outside $B_5(\rho) \setminus int(\Vec{L})$. The cutset edges are dashed and coloured in blue and the boundary $\partial \Vec{L}$ is dashed and orange. The interior $int(\vec{L})$ of the cutset is pictured on the right side.
  • Figure 4: Algorithmic construction of type-changing cutsets.

Theorems & Definitions (14)

  • Theorem 1
  • Remark 1
  • Proposition 1: Phase transition first-layer model
  • Lemma 1: Relation between the first- and second-layer measure
  • Remark 2
  • Definition 1
  • Remark 3
  • Definition 2
  • Lemma 2
  • proof
  • ...and 4 more