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UBC Theses and Dissertations

A macroscopic view of two discrete random models Alma Sarai, Hernandez Torres


This thesis investigates the large-scale behaviour emerging in two discrete models: the uniform spanning tree on ℤ³ and the chase-escape with death process. We consider the uniform spanning tree (UST) on ℤ³ as a measured, rooted real tree, continuously embedded into Euclidean space. The main result is on the existence of sub-sequential scaling limits and convergence under dyadic scalings. We study properties of the intrinsic distance and the measure of the sub-sequential scaling limits, and the behaviour of the random walk on the UST. An application of Wilson’s algorithm, used in the study of scaling limits, is also instrumental in a related problem. We show that the number of spanning clusters of the three-dimensional UST is tight under scalings of the lattice. Chase-escape is a competitive growth process in which red particles spread to adjacent uncoloured sites while blue particles overtake adjacent red particles. We propose a variant of the chase-escape process called chase-escape with death (CED). When the underlying graph of CED is a d-ary tree, we show the existence of critical parameters and characterize the phase transitions.

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