374 research outputs found
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
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 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
Information-Preserving Markov Aggregation
We present a sufficient condition for a non-injective function of a Markov
chain to be a second-order Markov chain with the same entropy rate as the
original chain. This permits an information-preserving state space reduction by
merging states or, equivalently, lossless compression of a Markov source on a
sample-by-sample basis. The cardinality of the reduced state space is bounded
from below by the node degrees of the transition graph associated with the
original Markov chain.
We also present an algorithm listing all possible information-preserving
state space reductions, for a given transition graph. We illustrate our results
by applying the algorithm to a bi-gram letter model of an English text.Comment: 7 pages, 3 figures, 2 table
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
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)
Advances in Industrial Crystallization
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K-independent percolation on trees
Consider the class of k-independent bond, respectively site, percolations
with parameter p on an infinite tree T. We derive tight bounds on p for both
a.s. percolation and a.s. nonpercolation. The bounds are continuous functions
of k and the branching number of T. This extends previous results by Lyons for
the independent case (k=0) and by Bollob\`as & Balister for 1-independent bond
percolations. Central to our argumentation are moment method bounds \`a la
Lyons supplemented by explicit percolation models \`a la Bollob\`as & Balister.
An indispensable tool is the minimality and explicit construction of Shearer's
measure on the k-fuzz of Z.Comment: 28 pages, 4 figure
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
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