| Summary: | In this thesis we study the contact process on various graphs. Chapter 1 introduces the contact process, and lists several known results about the process. Motivations of subsequent chapters are also given in Chapter 1. In Chapter 2 we consider an inhomogeneous variant of the classic contact process. In this variant each individual has fixed total outgoing infection rate, while among each edge the infection rate is non-constant. We also invent a corresponding branching random walk as a counterpart. We show that if the underlying graph is a supercritical Galton-Watson tree, then under certain circumstances, the branching random walk and the contact process exhibit a weak survival phase, that is, with positive probability the processes survive globally while die out locally almost surely. In Chapter 3 we consider the contact process on a random d-regular graph with size n, and we let n tend to infinity. Since the local geometry of a random regular graph resembles that of a regular tree, the appropriate analogue of the supercritical regime on the infinite graph is to choose the infection parameter λ > λ1(T d ), the lower critical value on the dregular tree. We show that in this case, for two typical vertices u, v on a typical graph G, a contact process initiated from u will infect v before time (C − ε) log n with vanishing probability, while it will infect v at time (C + ε) log n with a fixed positive probability. Also, we determine the density of the infection at time (C + ε) log n for a contact process started from the full occupancy state. In Chapter 4 a related problem of the supercritical contact process on a random regular graph is studied. We start the contact process from the full occupancy state. On a finite graph ∅ is always an absorbing state of the contact process, and the quantity of interest is the extinction time τn, the first time the contact process reaches ∅, on a graph of size n. We first establish an exponential bound on the extinction time, that is, with probability approaching 1, τn is at least exp(βn) for certain β > 0. Then, we show that the normalized extinction time, τn/EGτn, converges in law to a unit exponential distribution.
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