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UBC Theses and Dissertations

Coupling, matching, and equivariance Soo, Terry


This thesis consists of four research papers and one expository note that study factors of point processes in the contexts of thinning and matching. In "Poisson Splitting by Factors," we prove that given a Poisson point process on Rd with intensity λ, as a deterministic function of the process, we can colour the points red and blue, so that each colour class forms a Poisson point process on Rd, with any given pair of intensities summing λ; furthermore, the function can be chosen as an isometry-equivariant finitary factor (that is, if an isometry is applied to the points of the original process the points are still coloured the same way). Thus using only local information, without a central authority or additional randomization, points of a Poisson process can split into two groups, each of which are still Poisson. In "Deterministic Thinning of Finite Poisson Processes," we investigate similar questions for Poisson point processes on a finite volume. In this setting we find that even without considerations of equivariance, thinning can not always be achieved as a deterministic function of the Poisson process and the existence of such a function depends on the intensities of the original and resulting Poisson process. In "Insertion and Deletion Tolerance of Point Processes," we define for point processes a version of the concept of finite-energy. This simple concept has many interesting consequences. We explore the consequences of having finite-energy in the contexts of the Boolean continuum percolation model, Palm theory and stable matchings of point processes. In "Translation-Equivariant Matchings of Coin-Flips on Zd," as a factor of i.i.d. fair coin-flips on Zd, we construct perfect matchings of heads and tails and prove power law upper bounds on the expected distance between matched pairs. In the expository note "A Nonmeasurable Set from Coin-Flips," using the notion of an equivariant function, we give an example of a nonmeasurable set in the probability space for an infinite sequence of coin-flips.

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