Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps

Technical Summary: Uniqueness, Analyticity, and Mixing for Gibbs Point Processes via Spectral Gaps

Problem Statement
The paper addresses fundamental questions regarding the thermodynamic behavior of Gibbs point processes, which model particles interacting in the continuum via a pair potential $\phi$. The central challenges are establishing the uniqueness of infinite-volume Gibbs measures, proving the analyticity of the pressure (free energy), and demonstrating rapid mixing of associated Markov chains (sampling algorithms).

Historically, these properties have been proven using cluster expansion methods or Kirkwood–Salsburg equations, which typically yield uniqueness and analyticity only for activities $\lambda$ below a threshold of roughly $1/(e C_\phi)$, where $C_\phi$ is the temperedness constant. For the hard-sphere model, these classical bounds imply that the fluid regime (where the system behaves like a gas) exists only at very low densities, specifically $\lambda = \Theta(2^{-d})$ in high dimensions. The authors aim to improve these bounds significantly, particularly in high dimensions, and to identify potentials for which no phase transition occurs at any activity.

Methodology
The authors' approach diverges from traditional cluster expansions by leveraging spectral gap theory for continuous-time Markov chains. Their methodology proceeds through the following logical chain:

1. Spectral Gap Condition: The starting point is a result by Kondratiev, Kuna, and Ohlerich [39], which establishes that a specific positive-definiteness condition on an integral operator associated with the potential $\phi$ and activity $\lambda$ implies a spectral gap for spatial birth-death dynamics. The authors define the spectral threshold $\lambda_{\text{spec}}$ as the reciprocal of the supremum of the quadratic form associated with this operator.
2. From Spectral Gap to Spatial Mixing: The core technical contribution is proving that a uniform spectral gap for the birth-death dynamics implies Single-Site Strong Spatial Mixing (Single-Site SSM). Unlike previous works that required bounded-range potentials, the authors handle infinite-range potentials (decaying exponentially) by introducing "approximately local" functions and refining propagation-of-influence estimates for the birth-death semigroup.
3. From Spatial Mixing to Thermodynamic Properties:
   • Uniqueness: They show that Single-Site SSM implies the uniqueness of the infinite-volume Gibbs measure by summing the influence of boundary conditions and utilizing Poisson domination.
   • Analyticity: They prove that Single-Site SSM implies a zero-free region for the partition function in the complex plane. Using a Lee–Yang approach and an inductive argument on partition function ratios (inspired by Dobrushin and Shlosman but adapted for the continuum), they establish the analyticity of the infinite-volume pressure.
   • Mixing: They demonstrate that Single-Site SSM implies optimal mixing times ($O(\text{Vol}(\Lambda) \log \text{Vol}(\Lambda))$) for block dynamics (heat bath dynamics) using path coupling techniques adapted to the continuum.

Key Contributions and Results

• General Theorem (Theorem 1.1): For any repulsive potential $\phi$ with exponential decay, there is a unique infinite-volume Gibbs measure and the pressure is analytic for all activities $\lambda < \lambda_{\text{spec}}$. Furthermore, the system exhibits strong spatial mixing and optimal mixing times for block dynamics in this regime.
• Hard-Sphere Model Improvements:
  • The authors compute $\lambda_{\text{spec}}$ for the hard-sphere model in terms of Bessel functions.
  • High Dimensions: As $d \to \infty$, they show $\lambda_{\text{spec}} \approx (e/2)^{d/2} C_\phi^{-1}$. This represents an exponential improvement over previous bounds ($\lambda \sim 2^{-d}$).
  • Density Regime: In terms of expected density $\alpha$, the fluid regime is proven up to $\alpha = \Theta(d/2^d)$. This matches the density predicted by Parisi and Zamponi for the onset of slow mixing (up to a constant factor), whereas previous rigorous results were limited to $\Theta(2^{-d})$.
  • Low Dimensions: The bounds are also improved for fixed dimensions $d \ge 2$. For $d=2$, the bound is $C_\phi \lambda_{\text{spec}} \approx 7.56$, significantly higher than previous rigorous bounds (e.g., $3.233$ or percolation thresholds around $4.5$).
• Phase Transition-Free Potentials:
  • The authors exhibit repulsive, radial pair potentials for which $\lambda_{\text{spec}} = +\infty$. For these potentials, uniqueness and analyticity hold for all $\lambda > 0$, proving the existence of Gibbs point processes with no phase transition.
  • Specifically, they analyze the potential $\phi(x, y) = \log(1 - e^{-\|x-y\|^2})^{-1}$. While this potential has highly ordered ground states (the $E_8$ and Leech lattices in dimensions 8 and 24, respectively, as proven by Cohn et al.), the Gibbs process remains in a fluid phase for all activities.
• Gaussian Core Model: For the Gaussian core model with fixed inverse temperature $\beta$ and $d \to \infty$, the authors prove an exponential improvement in the uniqueness bound, showing $\lambda_{\text{spec}} \ge C_\beta 2^d / C_\phi$.
• Canonical Ensemble: The paper establishes the analyticity of the pressure in the canonical ensemble (fixed density) up to the density corresponding to $\lambda_{\text{spec}}$. For the hard-sphere model, this extends the analytic regime to densities of order $d 2^{-d}$.

Significance and Claims
The paper claims to provide the first rigorous proof of a Gibbs point process with a non-trivial, radial pair potential that exhibits no phase transition at any activity, despite having unique, highly ordered ground states. This challenges the intuition that ordered ground states necessarily lead to crystallization (phase transitions) at high densities.

Furthermore, the work provides the first asymptotically growing improvement over classical uniqueness and analyticity bounds for the hard-sphere model. The improvement grows exponentially with dimension, pushing the rigorous fluid regime to densities that are a factor of $d$ higher than previous results and matching the scaling predicted by physics literature for the onset of slow mixing.

The authors emphasize that their results unify several "fluid-like" properties (uniqueness, analyticity, spatial mixing, temporal mixing, and efficient sampling) under a single spectral gap condition, extending the scope of these implications from bounded-range lattice models to continuum models with unbounded-range potentials.