Perfect simulation of some spatial point processes

University dissertation from Göteborg : Göteborgs universitet

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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