scholarly journals Subcritical phase of $d$-dimensional Poisson–Boolean percolation and its vacant set

2020 ◽  
Vol 3 ◽  
pp. 677-700 ◽  
Author(s):  
Hugo Duminil-Copin ◽  
Aran Raoufi ◽  
Vincent Tassion
Keyword(s):  
Boolean Percolation

Boolean Percolation on Doubling Graphs

2019 ◽  
Vol 178 (3) ◽  
pp. 814-831
Author(s):  
Cristian F. Coletti ◽  
Daniel Miranda ◽  
Sebastian P. Grynberg
Keyword(s):  
Boolean Percolation

scholarly journals Cluster Expansions for GIBBS Point Processes

2019 ◽  
Vol 51 (4) ◽  
pp. 1129-1178 ◽  
Author(s):  
S. Jansen
Keyword(s):  
Correlation Functions ◽  
Point Processes ◽  
Branching Processes ◽  
Factorial Moment ◽  
Convergence Condition ◽  
Random Measures ◽  
Cluster Expansions ◽  
Gibbs Point Processes ◽  
Random Connection Model ◽  
Boolean Percolation

AbstractWe provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood–Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández, and Procacci (2010). In addition, we provide formulas for cumulants of double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

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