UBC Theses and Dissertations
Geometry of random spaces : geodesics and susceptibility Kolesnik, Brett Thomas
This thesis investigates the geometry of random spaces. Geodesics in random surfaces. The Brownian map, developed by Le Gall and Miermont, is a random metric space arising as the scaling limit of random planar maps. Its construction involves Aldous’ continuum random tree, the canonical random real tree, and Brownian motion, an almost surely continuous but nowhere differentiable path. As a result, the Brownian map is a non-differentiable surface with a fractal geometry that is much closer to that of a real tree than a smooth surface. A key feature, observed by Le Gall, is the confluence of geodesics phenomenon, which states that any two geodesics to a typical point coalesce before reaching the point. We show that, in fact, geodesics to anywhere near a typical point pass through a common confluence point. This leads to information about special points that had remained largely mysterious. Our main result is the almost everywhere continuity and uniform stability of the cut locus of the Brownian map. We also classify geodesic networks that are dense and find the Hausdorff dimension of the set of pairs that are joined by each type of network. Susceptibility of random graphs. Given a graph G=(V,E) and an initial set I of active vertices in V, the r-neighbour bootstrap percolation process, attributed to Chalupa, Leath and Reich, is a cellular automaton that evolves by activating vertices with at least r active neighbours. If all vertices in V are activated eventually, we say that I is contagious. A graph with a small contagious set is called susceptible. Bootstrap percolation has been analyzed on several deterministic graphs, such as grids, lattices and trees. More recent work studies the model on random graphs, such as the fundamental Erdős–Rényi graph G(n,p). We identify thresholds for the susceptibility of G(n,p), refining approximations by Feige, Krivelevich and Reichman. Along the way, we obtain large deviation estimates that complement central limit theorems of Janson, Łuczak, Turova and Vallier. We also study graph bootstrap percolation, a variation due to Bollobás. Our main result identifies the sharp threshold for K4-percolation, solving a problem of Balogh, Bollobás and Morris.
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