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

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.

💡 Research Summary

The paper investigates first‑order phase transitions in continuum Gibbs point processes whose interactions satisfy a “saturation” property. In such models the local energy in a region of high particle density depends only on the number of points inside that region, i.e. the energy is linear in the local particle count once the configuration is locally homogeneous. The authors build on the Pirogov‑Sinai‑Zahradník (PSZ) theory, originally developed for lattice spin systems, and adapt it to the continuum setting by means of a coarse‑graining procedure that partitions ℝⁿ into cubic tiles of side δ.

The main contributions are twofold. First, a general theorem (Theorem 1) is proved: for any Hamiltonian that fulfills a set of structural assumptions (stability, finite range, translation invariance, and especially the saturation condition (E5)), there exist two distinct infinite‑volume Gibbs measures P⁺ and P⁻ at suitable values of the activity z and inverse temperature β. These measures have different particle intensities ρ(P⁺) > ρ(P⁻), and the pressure ψ(z,β) is non‑differentiable, which is the hallmark of a first‑order transition. The proof proceeds by (i) defining a contour representation of deviations from the two pure phases (dense and empty), (ii) establishing a Peierls bound that guarantees an exponential penalty proportional to the contour size, (iii) performing a polymer expansion over compatible contours, and (iv) showing convergence of the expansion for sufficiently large z and β. The truncated pressures associated with each phase are shown to be distinct, yielding the pressure jump.

Second, the authors introduce a new class of “diluted pairwise interactions”. Starting from a radial pair potential φ with compact support, they define a dilution scale R>0 and consider only interactions between points that lie within distance R of each other. The Hamiltonian is the double integral of φ over the R‑neighbourhood of the configuration. By averaging this interaction over each tile they obtain a tile‑energy E₀ that satisfies the saturation condition when the tile size δ and the interaction range R are chosen appropriately (δ ≤ R√d, L > R₂+2√d δ). Consequently, the general theorem applies, establishing a first‑order transition for the diluted model. As a corollary, any strongly repulsive short‑range potential (e.g., truncated Lennard‑Jones) can be truncated at the origin and still exhibit a first‑order transition for any dilution scale, providing a new route to prove phase transitions for pairwise potentials that were previously intractable.

The paper also revisits two classical examples that fit the framework: (a) the K‑nearest‑neighbour Strauss interaction, where each point interacts only with its K closest neighbours within a fixed radius, and (b) the Quermass interaction with random bounded radii, which is a morphological model based on Minkowski functionals (volume, surface area, Euler characteristic). In both cases the authors verify the saturation property and the auxiliary bounds (E3)–(E4).

Technical highlights include:

• A precise definition of homogeneous dense (Ω⁽¹⁾) and empty (Ω⁽⁰⁾) configurations around a tile, and the construction of the energy function E₀ that depends only on the number of points in the tile for such configurations.
• The use of Dobrushin‑Lanford‑Ruelle (DLR) equations to characterize Gibbs measures, together with existence results from earlier work (Dereudre 2019, Rœlly‑Zass 2020).
• A contour formalism that treats deviations from the pure phases as polymers, with activities bounded by exp(−βκ|Γ|) where κ>0 follows from the saturation‑induced Peierls bound.
• A careful truncation of the pressure to control contributions from large contours, leading to two analytically tractable pressure functions ψ⁺ and ψ⁻.
• Demonstration that the diluted pairwise interaction satisfies all required assumptions, thereby extending the PSZ‑type analysis to a broader class of continuum models.

Overall, the work provides a robust, model‑independent methodology for proving first‑order phase transitions in a wide variety of continuum Gibbs point processes. By translating the lattice‑based PSZ machinery to the continuum via coarse‑graining and saturation, the authors open the door to studying phase coexistence in more realistic particle systems, including those with complex morphological interactions or renormalised pairwise potentials. Future directions suggested include relaxing the saturation condition, exploring non‑cubic coarse‑graining, and extending the analysis to dynamical (Markov) point processes or systems with additional internal degrees of freedom.