Gap metrics for stationary point processes and quantitative convexity of the free energy

2025/09/10 by Huesmann, Martin, Müller, Bastian · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.08659

Abstract

In this article, we are interested in convexity properties of the free energy for stationary point processes on \mathbb R w.r.t. a new geometry inspired by optimal transport. We will show for a rich class of pairwise interaction energies A) quantified strict convexity of the free energy implying uniqueness of minimizers B) existence of a gradient flow curve of the free energy w.r.t. the new metric converging exponentially fast to the unique minimizer. Examples for energies for which A holds include logarithmic or Riesz interactions with parameter 0

Citations

• Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes
• The variational principle for a marked Gibbs point process with infinite-range multibody interactions
• (Non)-hyperuniformity of perturbed lattices
• The link between hyperuniformity, Coulomb energy, and Wasserstein distance to Lebesgue for two-dimensional point processes
• On the Wasserstein distance between a hyperuniform point process and its mean
• Hyperuniformity and optimal transport of point processes
• A Benamou-Brenier formula for transport distances between stationary random measures
• Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy
• Annealed quantitative estimates for the quadratic 2D-discrete random matching problem
• Curvature bound of Dyson Brownian Motion
• Decay of correlations and thermodynamic limit for the circular Riesz gas
• A fluctuation result for the displacement in the optimal matching problem
• Minimal matchings of point processes
• The one-dimensional log-gas free energy has a unique minimiser
• Microscopic description of Log and Coulomb gases
• Introduction to the theory of Gibbs point processes
• A PDE approach to a 2-dimensional matching problem
• Variational principle for Gibbs point processes with finite range interaction
• Large Deviation Principle for Empirical Fields of Log and Riesz Gases
• Infinite-dimensional stochastic differential equations related to random\n matrices
• Poisson Matching
• Extra heads and invariant allocations

Cited by

• Next-order asymptotics for the volume of Schatten balls

Related

• Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy
• Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes
• Strict convexity of the free energy for non-convex gradient models at moderate β
• Geometric inequalities via a general comparison principle for interacting gases
• The one-dimensional log-gas free energy has a unique minimiser