388 research outputs found
Shearer's point process, the hard-sphere model and a continuum Lov\'asz Local Lemma
A point process is R-dependent, if it behaves independently beyond the
minimum distance R. This work investigates uniform positive lower bounds on the
avoidance functions of R-dependent simple point processes with a common
intensity. Intensities with such bounds are described by the existence of
Shearer's point process, the unique R-dependent and R-hard-core point process
with a given intensity. This work also presents several extensions of the
Lov\'asz Local Lemma, a sufficient condition on the intensity and R to
guarantee the existence of Shearer's point process and exponential lower
bounds. Shearer's point process shares combinatorial structure with the
hard-sphere model with radius R, the unique R-hard-core Markov point process.
Bounds from the Lov\'asz 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 on the uniqueness of the Gibbs measure
of the hard-sphere model via an inductive approach \`a la Dobrushin
Disagreement percolation for the hard-sphere model
Disagreement percolation connects a Gibbs lattice gas and i.i.d. site
percolation on the same lattice such that non-percolation implies uniqueness of
the Gibbs measure. This work generalises disagreement percolation to the
hard-sphere model and the Boolean model. Non-percolation of the Boolean model
implies the uniqueness of the Gibbs measure and exponential decay of pair
correlations and finite volume errors. Hence, lower bounds on the critical
intensity for percolation of the Boolean model imply lower bounds on the
critical activity for a (potential) phase transition. These lower bounds
improve upon known bounds obtained by cluster expansion techniques. The proof
uses a novel dependent thinning from a Poisson point process to the hard-sphere
model, with the thinning probability related to a derivative of the free
energy
Disagreement percolation for Gibbs ball models
We generalise disagreement percolation to Gibbs point processes of balls with
varying radii. This allows to establish the uniqueness of the Gibbs measure and
exponential decay of pair correlations in the low activity regime by comparison
with a sub-critical Boolean model. Applications to the Continuum Random Cluster
model and the Quermass-interaction model are presented. At the core of our
proof lies an explicit dependent thinning from a Poisson point process to a
dominated Gibbs point process.Comment: 23 pages, 0 figure Correction, from the published version, of the
proof of Section
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
- …