Percolation of Hard Disks

Random arrangements of points in the plane, interacting only through a simple hard-core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that, at high intensity, an infinite connected cluster of excluded volume appears almost surely.

Download Full-text

Likelihood-based inference for Matérn type-III repulsive point processes

Advances in Applied Probability ◽

10.1239/aap/1261669580 ◽

2009 ◽

Vol 41 (4) ◽

pp. 958-977 ◽

Cited By ~ 12

Author(s):

Mark L. Huber ◽

Robert L. Wolpert

Keyword(s):

Point Process ◽

Poisson Point Process ◽

Point Processes ◽

Simulation Method ◽

Perfect Simulation ◽

Accurate Approximation ◽

Intensity Parameter ◽

Type Iii ◽

Competition For Resources ◽

As If

In a repulsive point process, points act as if they are repelling one another, leading to underdispersed configurations when compared to a standard Poisson point process. Such models are useful when competition for resources exists, as in the locations of towns and trees. Bertil Matérn introduced three models for repulsive point processes, referred to as types I, II, and III. Matérn used types I and II, and regarded type III as intractable. In this paper an algorithm is developed that allows for arbitrarily accurate approximation of the likelihood for data modeled by the Matérn type-III process. This method relies on a perfect simulation method that is shown to be fast in practice, generating samples in time that grows nearly linearly in the intensity parameter of the model, while the running times for more naive methods grow exponentially.

Download Full-text

Likelihood-based inference for Matérn type-III repulsive point processes

Advances in Applied Probability ◽

10.1017/s0001867800003682 ◽

2009 ◽

Vol 41 (04) ◽

pp. 958-977 ◽

Cited By ~ 5

Author(s):

Mark L. Huber ◽

Robert L. Wolpert

Keyword(s):

Point Process ◽

Poisson Point Process ◽

Point Processes ◽

Simulation Method ◽

Perfect Simulation ◽

Accurate Approximation ◽

Intensity Parameter ◽

Type Iii ◽

Competition For Resources ◽

As If

In a repulsive point process, points act as if they are repelling one another, leading to underdispersed configurations when compared to a standard Poisson point process. Such models are useful when competition for resources exists, as in the locations of towns and trees. Bertil Matérn introduced three models for repulsive point processes, referred to as types I, II, and III. Matérn used types I and II, and regarded type III as intractable. In this paper an algorithm is developed that allows for arbitrarily accurate approximation of the likelihood for data modeled by the Matérn type-III process. This method relies on a perfect simulation method that is shown to be fast in practice, generating samples in time that grows nearly linearly in the intensity parameter of the model, while the running times for more naive methods grow exponentially.

Download Full-text

A generalization of Matérn hard-core processes with applications to max-stable processes

Journal of Applied Probability ◽

10.1017/jpr.2020.66 ◽

2020 ◽

Vol 57 (4) ◽

pp. 1298-1312

Author(s):

Martin Dirrler ◽

Christopher Dörr ◽

Martin Schlather

Keyword(s):

Point Process ◽

Point Processes ◽

Hard Core ◽

Domain Of Attraction ◽

Stable Processes ◽

Poisson Point Processes ◽

Original Process ◽

Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

Download Full-text