Random Spatial Structures and Sums

Abstract

The thesis contains three articles about three different models, all of which are about probability in infinite spaces. It begins with an informal introductory chapter accessible to a general reader who feels comfortable with mathematics. In the second chapter we consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or -1, with probability 1-p. We ask whether an infinite path pi exists, with the property that the partial sums of the +1s along pi are uniformly bounded, and whether there exists an infinite path pi with the property that the partial sums along pi are equal to zero infinitely often. The answers to these questions depend on the type of path one allows, the value of p and the uniform bound specified. We show that phase transitions occur for these phenomena. Moreover, we make a surprising connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each vertex at most finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson process in the plane. For the recurrence problem, we also show that the probability of finding such a path is monotone in p, for p > ½. Continuum percolation models in which each point of a two-dimensional Poisson point process is the centre of a disc of given (or random) radius r, have been extensively studied. In the third chapter, we consider the generalisation in which a deterministic algorithm (given the points of the point process) places the discs on the plane, in such a way that each disc covers at least one point of the point process and that each point is covered by at least one disc. This gives a model for wireless communication networks, which was the original motivation to study this class of problems. We look at the percolation properties of this generalised model, showing that an unbounded connected component of discs does not exist, almost surely, for small values of the density lambda of the Poisson point process, for any covering algorithm. In general, it turns out not to be true that unbounded connected components arise when lambda is taken sufficiently high, even when we require the algorithm to be invariant under shifts of the points. However, we identify some large families of covering algorithms, for which such an unbounded component does arise for large values of lambda We show how a simple scaling operation can change the percolation properties of the model, leading to the almost sure existence of an unbounded connected component for large values of lambda , for any covering algorithm. Finally, we show that a large class of covering algorithms, that arise in many practical applications, can get arbitrarily close to achieving a minimal density of covering discs. We also show (constructively) the existence of algorithms that achieve this minimal density. In the fourth chapter, we construct certain particle systems on Z-exp.Z-exp.d. These systems start in configurations given by a stationary, ergodic law, such that the expectation of the absolute number of particles per site is finite. Particles are added and removed at rates which are functions of the configuration of particles. These functions are of bounded variation and are such that the rate at which particles are added at each is site is almost surely finite. Negative numbers of particles are allowed.