3 research outputs found
Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma
A point process is R-dependent if it behaves independently beyond the minimum
distance R. In this paper we investigate uniform positive lower bounds on the avoidance
functions of R-dependent simple point processes with a common intensity. Intensities
with such bounds are characterised by the existence of Shearer’s point process, the unique
R-dependent and R-hard-core point process with a given intensity. We also present
several extensions of the Lovász local lemma, a sufficient condition on the intensity
andR to guarantee the existence of Shearer’s point process and exponential lower bounds.
Shearer’s point process shares a combinatorial structure with the hard-sphere model with
radius R, the unique R-hard-core Markov point process. Bounds from the Lovász local
lemma convert into lower bounds on the radius of convergence of a high-temperature
cluster expansion of the hard-sphere model. This recovers a classic result of Ruelle
(1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive
approach of Dobrushin (1996)
Clique trees of infinite locally finite chordal graphs
We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting
Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics
We study a stationary Gibbs particle process with deterministically bounded particles on
Euclidean space defined in terms of an activity parameter and non-negative interaction
potentials of finite range. Using disagreement percolation we prove exponential decay of
the correlation functions, provided a dominating Boolean model is subcritical. We also
prove this property for the weighted moments of a U-statistic of the process. Under the
assumption of a suitable lower bound on the variance, this implies a central limit theorem
for such U-statistics of the Gibbs particle process. A byproduct of our approach is a new
uniqueness result for Gibbs particle processes