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Potential-weighted connective constants and uniqueness of Gibbs measures

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 Added by Will Perkins
 Publication date 2021
  fields Physics
and research's language is English




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We define a 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 $mathbb R^d$ 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.



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The connective constant $mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $bullet$ We present upper and lower bounds for $mu$ in terms of the vertex-degree and girth of a transitive graph. $bullet$ We discuss the question of whether $mugephi$ for transitive cubic graphs (where $phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $bullet$ We present strict inequalities for the connective constants $mu(G)$ of transitive graphs $G$, as $G$ varies. $bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $bullet$ We describe so-called graph height functions within an account of bridges for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $bullet$ The review closes with a brief account of the speed of SAW.
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