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<p>We define a new ‘potential-weighted connective constant’ that measures the effective strength of a repulsive pair potential of a Gibbs point process modulated by the geometry of the underlying space. We then show that this definition leads to improved bounds for Gibbs uniqueness for all non-trivial repulsive pair potentials on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2024_5214_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {R}}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> and other metric measure spaces. We do this by constructing a tree-branching collection of densities associated to the point process that captures the interplay between the potential and the geometry of the space. When the activity is small as a function of the potential-weighted connective constant this object exhibits an infinite-volume uniqueness property. On the other hand, we show that our uniqueness bound can be tight for certain spaces: the same infinite-volume object exhibits non-uniqueness for activities above our bound in the case when the underlying space has the geometry of a tree.</p>

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Potential-Weighted Connective Constants and Uniqueness of Gibbs Measures

  • 期刊论文

  • 发表日期：

    2025年1月11日

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  • Marcus Michelen,
  • Will Perkins

摘要

We define a new ‘potential-weighted connective constant’ that measures the effective strength of a repulsive pair potential of a Gibbs point process modulated by the geometry of the underlying space. We then show that this definition leads to improved bounds for Gibbs uniqueness for all non-trivial repulsive pair potentials on \({\mathbb {R}}^d\) R d and other metric measure spaces. We do this by constructing a tree-branching collection of densities associated to the point process that captures the interplay between the potential and the geometry of the space. When the activity is small as a function of the potential-weighted connective constant this object exhibits an infinite-volume uniqueness property. On the other hand, we show that our uniqueness bound can be tight for certain spaces: the same infinite-volume object exhibits non-uniqueness for activities above our bound in the case when the underlying space has the geometry of a tree.