Perfect simulation of some spatial point processes
Abstract: Coupling from the past (CFTP) algorithms are presented that generate perfectly distributed samples from the multi-type Widom--Rowlinson (W--R) model and some generalizations of it. The classical W--R model is a point process in the plane or the space consisting of points of several different types. Points of different types are not allowed to be closer than some specified distance, whereas points of the same type can be arbitrary close. An application can be to describe certain gases consisting of several types of particles. We also consider a soft-core W--R model, where points of different types are not completely forbidden to be close to each other, just inhibited in various degrees. Furthermore, we allow the hindrance between two points of different types to be explained by more than the Euclidean distance between them. In particular we consider a stick-model where the hindrance is defined by imaginary sticks, with centers at the associated points, and where sticks are not allowed to cross each other. The different directions of the sticks (a finite number), represent the different types of the points. A CFTP algorithm is also given for a soft-core version of the stick-model. Simulation studies indicate that the runtime of the CFTP algorithm for the multi-type W--R model in the symmetric case (i.e.\ equal intensities), first grows exponentially with the intensity, but then suddenly, when the intensity becomes larger seems to be superexponential. This change in growth may be explained by a phase transition. We also present a CFTP algorithm that yields samples without edge effects from the multi-type W--R model. The underlying idea behind this algorithm is to not only simulate backwards in time, but also outwards in space. This algorithm does not always terminate for large intensities of the points. A bound on sufficiently small intensities for the algorithm to terminate is given.
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