scholarly journals Continuum percolation thresholds in two dimensions

2012 ◽  
Vol 86 (6) ◽  
Author(s):  
Stephan Mertens ◽  
Cristopher Moore
Keyword(s):  
Two Dimensions ◽  
Continuum Percolation ◽  
Percolation Thresholds

scholarly journals Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

2013 ◽  
Vol 45 (1) ◽  
pp. 20-36 ◽  
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
V. Schmidt
Keyword(s):  
Point Process ◽  
Open Problem ◽  
Poisson Point Process ◽  
Point Processes ◽  
Geometric Graphs ◽  
Continuum Percolation ◽  
Random Geometric Graphs ◽  
Spanning Forests ◽  
Percolation Thresholds ◽  
Relative Neighborhood Graph

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point process X ⊂ ℝd is an open problem for dimension d>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). For n=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity of Gn(X) holds for all n≥2, all dimensions d≥2, and also point processes X more general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surely X does not admit generalized descending chains.

scholarly journals Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

2013 ◽  
Vol 45 (01) ◽  
pp. 20-36 ◽  
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
V. Schmidt
Keyword(s):  
Point Process ◽  
Open Problem ◽  
Poisson Point Process ◽  
Point Processes ◽  
Geometric Graphs ◽  
Continuum Percolation ◽  
Random Geometric Graphs ◽  
Spanning Forests ◽  
Percolation Thresholds ◽  
Relative Neighborhood Graph

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point processX⊂ ℝdis an open problem for dimensiond>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). Forn=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity ofGn(X) holds for alln≥2, all dimensionsd≥2, and also point processesXmore general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surelyXdoes not admit generalized descending chains.

scholarly journals Exact bond percolation thresholds in two dimensions

2006 ◽  
Vol 39 (49) ◽  
pp. 15083-15090 ◽  
Author(s):  
Robert M Ziff ◽  
Christian R Scullard
Keyword(s):  
Two Dimensions ◽  
Bond Percolation ◽  
Percolation Thresholds
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