High-dimensional Poisson–Voronoi geometry and threshold phenomena
This thesis studies random geometric structures and their phase transitions. These are sudden changes in behaviour when a parameter reaches a critical value. High dimensions play a central role. Familiar geometric intuition often becomes unreliable there, while many random quantities become surprisingly predictable.
The first part considers percolation in Poisson–Voronoi tessellations and the related Gabriel graph. Each cell or vertex is independently declared open, and we ask when an infinite connected cluster can appear. In high dimensions, the threshold is largely determined by the expected number of neighbours of a typical vertex.
The second part studies the random Borsuk graph. Random points are placed on a sphere and nearly opposite points are connected. We ask when the graph needs more colours. For several transitions in the chromatic number, we identify sharp thresholds. A small change in the connection range can lead to a large change in behaviour.
The final part returns to the Poisson–Voronoi tessellation and examines the shape of a typical cell. After a natural rescaling, the radii of its largest inscribed and smallest enclosing balls, its diameter and its average width become almost deterministic. Yet the cell does not turn into a smooth ball. Its polyhedral structure remains visible.
The common thread is that high dimensions bring both regularity and complexity. Global properties become predictable, while local geometry still creates varied and sometimes surprising phase transitions.