Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Roy, Rahul (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

The model of continuum percolation can be described as follows. We start with a homogeneous Poisson point process X. At each point of X we centre a ball with a random radius such that the radii corresponding to different points are independent of each other and also independent of the Poisson process X. In this way, the space is divided into two regiorns, the covered region or the occupied region consisting of the region which is covered by at least one ball, and the uncovered region or the vacant region which is complement of the covered region. In this dissertation we study various properties of this covered region.Percolation theory first found its application in solid-state physics, but in the later years it has been applied in many more diverse fields like geophysics, astrophysics, chernistry of polymers etc. In physics, the phenomenon of phase transition as observed in stirred mixtures of immiscible liquids is modelled by the continuum percolation model. Consider the following experiment of adding oil slowly in water and stirring it constantly. If the amount of oil added, i.e. fraction of oil to water, is very small, droplets of random size of oil are formed in the background of water. If we keep on adding more oil, the system goes into phase change to reach a situation where water droplets are dispersed in oil. The physical literature on this subject is primarily a study based on Monte Carlo simulations although heuristic arguments are provided in some of the works (see, for example, Scher and Zallen [1970), Pike and Seager (1974), Kertesz and Vicsek (1982), Gawlinski and Redner [1983), Phani and Dhar (1984]).The mathematical study of continuum percolation was initiated by Hall [1985, 1986). This model which is known as the Boolean model in stochastic geometry, has been studied extensively by geometers, albeit with a view of solving problems of a geometric nature. Hall (1988] is an excellent book devoted to the study of the geometric and statistical aspects of the Boolean model. The model of continuum percolation was first introduced in a study of communication networks by Gilbert [1961} as a model for the growth and structure of random networks. Men- shikov, Molchanov and Sidorenko (1985), Zuev and Sidorenko [1985), Menshikov (1986), Roy (1990), Alexander [1993), Meester and Roy (1994] studied the model to obtain various results.The other model of continuous percolation that we study is the * random connection model. Given a homogeneous Poisson point process X, another way of constructing random objects is to connect the pair of points according to a given rule. In the random connection model, we connect a pair of points z1, z2 with the probability g(jz, - 12l) where g is a given function known as the connection function and | ·| is Euclidean distance. The components here are defined in the usual w. The transmission of disease among trees in a forest can be modelled by such a process. Penrose (1991], Burton and Meester [1993), Meester [1994] studied this model to obtain various results. In, the next section we introduce the models and give the necess. definitions and results.


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