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

Martin Huesmann; Bastian Müller

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arXiv · 2509.08659

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

Martin Huesmann·Bastian Müller

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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 $0<s<1$, examples for which A and B hold are hypersingular Riesz or Yukawa interactions.

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Martin Huesmann, Bastian Müller. 2025-09-10. Gap metrics for stationary point processes and quantitative convexity of the free energy. https://arxiv.org/abs/2509.08659

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