scholarly journals Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma

2017 ◽  
Vol 49 (1) ◽  
pp. 1-23
Author(s):  
Christoph Hofer-Temmel
Keyword(s):  
Point Process ◽  
Lower Bounds ◽  
Hard Sphere ◽  
Hard Core ◽  
Combinatorial Structure ◽  
Hard Sphere Model ◽  
Sphere Model ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Lovasz Local Lemma

AbstractA point process isR-dependent if it behaves independently beyond the minimum distanceR. In this paper we investigate uniform positive lower bounds on the avoidance functions ofR-dependent simple point processes with a common intensity. Intensities with such bounds are characterised by the existence of Shearer's point process, the uniqueR-dependent andR-hard-core point process with a given intensity. We also present several extensions of the Lovász local lemma, a sufficient condition on the intensity andRto 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 radiusR, the uniqueR-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).

scholarly journals An Improvement of the Lovász Local Lemma via Cluster Expansion

2011 ◽  
Vol 20 (5) ◽  
pp. 709-719 ◽  
Author(s):  
RODRIGO BISSACOT ◽  
ROBERTO FERNÁNDEZ ◽  
ALDO PROCACCI ◽  
BENEDETTO SCOPPOLA
Keyword(s):  
Partition Function ◽  
Recent Result ◽  
Cluster Expansion ◽  
Lattice Gas ◽  
Hard Core ◽  
Independent Set ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Lovasz Local Lemma

An old result by Shearer relates the Lovász local lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard-core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lovász local lemma. As an application we obtain tighter bounds on conditions for the existence of Latin transversal matrices.

scholarly journals Using Lovász Local Lemma in the Space of Random Injections

10.37236/981 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Linyuan Lu ◽  
László Székely
Keyword(s):  
Lower Bounds ◽  
Extremal Problems ◽  
Existence Results ◽  
Dependency Graph ◽  
Asymptotic Enumeration ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Enumeration Problems ◽  
Lovasz Local Lemma

The Lovász Local Lemma is known to have an extension for cases where independence is missing but negative dependencies are under control. We show that this is often the case for random injections, and we provide easy-to-check conditions for the non-trivial task of verifying a negative dependency graph for random injections. As an application, we prove existence results for hypergraph packing and Turán type extremal problems. A more surprising application is that tight asymptotic lower bounds can be obtained for asymptotic enumeration problems using the Lovász Local Lemma.

scholarly journals Lower Bounds on van der Waerden Numbers: Randomized- and Deterministic-Constructive

10.37236/551 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
William Gasarch ◽  
Bernhard Haeupler
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Efficient Algorithm ◽  
Probabilistic Method ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Constructive Proofs ◽  
Different Types ◽  
Lovasz Local Lemma

The van der Waerden number $W(k,2)$ is the smallest integer $n$ such that every $2$-coloring of 1 to $n$ has a monochromatic arithmetic progression of length $k$. The existence of such an $n$ for any $k$ is due to van der Waerden but known upper bounds on $W(k,2)$ are enormous. Much effort was put into developing lower bounds on $W(k,2)$. Most of these lower bound proofs employ the probabilistic method often in combination with the Lovász Local Lemma. While these proofs show the existence of a $2$-coloring that has no monochromatic arithmetic progression of length $k$ they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on $W(k,2)$ in this light. We show how known nonconstructive lower bound proofs based on the Lovász Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms.

scholarly journals Descending chains, the lilypond model, and mutual-nearest-neighbour matching

2005 ◽  
Vol 37 (03) ◽  
pp. 604-628 ◽  
Author(s):  
Daryl J. Daley ◽  
Günter Last
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Hard Sphere ◽  
Significant Part ◽  
Hard Sphere Model ◽  
Nearest Neighbour ◽  
Sphere Model ◽  
Stationary Point Process ◽  
Unit Speed ◽  
General Conditions

We consider a hard-sphere model in ℝdgenerated by a stationary point processNand thelilypond growth protocol: at time 0, every point ofNstarts growing with unit speed in all directions to form a system of balls in which any particular ball ceases its growth at the instant that it collides with another ball. Some quite general conditions are given, under which it is shown that the model is well defined and exhibits no percolation. The absence of percolation is attributable to the fact that, under our assumptions, there can be nodescending chainsinN. The proof of this fact forms a significant part of the paper. It is also shown that, in the absence of descending chains, mutual-nearest-neighbour matching can be used to construct a bijective point map as defined by Thorisson.

scholarly journals Descending chains, the lilypond model, and mutual-nearest-neighbour matching

2005 ◽  
Vol 37 (3) ◽  
pp. 604-628 ◽  
Author(s):  
Daryl J. Daley ◽  
Günter Last
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Hard Sphere ◽  
Significant Part ◽  
Hard Sphere Model ◽  
Nearest Neighbour ◽  
Sphere Model ◽  
Stationary Point Process ◽  
Unit Speed ◽  
General Conditions

We consider a hard-sphere model in ℝd generated by a stationary point process N and the lilypond growth protocol: at time 0, every point of N starts growing with unit speed in all directions to form a system of balls in which any particular ball ceases its growth at the instant that it collides with another ball. Some quite general conditions are given, under which it is shown that the model is well defined and exhibits no percolation. The absence of percolation is attributable to the fact that, under our assumptions, there can be no descending chains in N. The proof of this fact forms a significant part of the paper. It is also shown that, in the absence of descending chains, mutual-nearest-neighbour matching can be used to construct a bijective point map as defined by Thorisson.

Sign in / Sign up

Export Citation Format

Share Document