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First-order phase transition for Gibbs point processes with saturated interactions

David Dereudre, Christopher Renaud-Chan

Abstract

We study first-order phase transitions in continuum Gibbs point processes with saturated interactions. These interactions form a broad class of Hamiltonians in which the local energy in regions of high particle density depends only on the number of points. Building on ideas of Pirogov-Sinai-Zahradnik theory and its adaptations to the continuum, we develop a general method for establishing the existence of two distinct infinite-volume Gibbs measures with different intensities in this setting, demonstrating a first-order phase transition. Our approach extends previous results obtained for the Quermass model and applies in particular to a new class of diluted pairwise interactions introduced in this work.

First-order phase transition for Gibbs point processes with saturated interactions

Abstract

We study first-order phase transitions in continuum Gibbs point processes with saturated interactions. These interactions form a broad class of Hamiltonians in which the local energy in regions of high particle density depends only on the number of points. Building on ideas of Pirogov-Sinai-Zahradnik theory and its adaptations to the continuum, we develop a general method for establishing the existence of two distinct infinite-volume Gibbs measures with different intensities in this setting, demonstrating a first-order phase transition. Our approach extends previous results obtained for the Quermass model and applies in particular to a new class of diluted pairwise interactions introduced in this work.
Paper Structure (14 sections, 12 theorems, 138 equations, 2 figures)

This paper contains 14 sections, 12 theorems, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and models
  3. Setting and notations
  4. Interaction
  5. Coarse graining and saturation
  6. Gibbs marked point process
  7. Contours
  8. Results
  9. Proof of Theorem \ref{['thm: liquid_gaz1']}
  10. Existence of Gibbs point processes for different boundary conditions
  11. Polymer development
  12. Truncated pressures and weights
  13. Proof of Theorem \ref{['thm: liquid_gaz1']}
  14. Proof of Theorem \ref{['thm:fopt_diluted_pair']}

Key Result

Theorem 1

Let $H$ be a Hamiltonian that satisfies H:stability-H:finiterange such that $E_0$ exists and satisfies E:non-degenerate-E:saturation. Furthermore, we suppose that the interaction satisfies a Peierls-like condition, i.e. there is $\overline{e}_+>0$ such that for any contour $\gamma$ and any configura For any $\beta>0$, we fix $z_\beta^-$ and $z_\beta^+$ as and the open interval $O_\beta := (z_\bet

Figures (2)

  • Figure 1: The contour corresponds to the grey areas, while the blue and red squares represent the tiles at the boundary of the contour where the spins are $\sharp$ and $1-\sharp$, respectively. In Figure \ref{['fig_a']}, the contour $\Gamma =\{\gamma_1, \gamma_2\}$ is achievable by some configuration $\omega$ because the label of $A$ matches for both $\gamma_1$ and $\gamma_2$. In contrast, in Figure \ref{['fig_b']}, the contour $\Gamma = \{\gamma_1, \gamma_2\}$ is not globally achievable by any configuration. In this case, $\Gamma \in \mathcal{C}^{{{{{(\sharp)}}}}}(\Lambda)$, where the types of $\gamma_1$ and $\gamma_2$ are the same, but the labels of $A$ are mismatched. (reproduced from renaudchanQuermass)
  • Figure 2: $B_{sec}(z,x)$

Theorems & Definitions (19)

  • Definition 2.1
  • Theorem 1
  • Corollary 3.1
  • Theorem 2
  • Corollary 3.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • ...and 9 more