Bootstrap Percolation of Holes Distributed by a Non-Space Homogeneous Poisson Process.

Sammanfattning

In the natural sciences, there are many processes involving the loss of tension in a stretched membrane. Thus, one can ask what would happen to a stretched membrane from which certain parts have been removed. One would see that overlapping parts result in some region of the membrane losing tension. Then, if one considers any part that loses tension to also be removed from the membrane, this newly removed section could intersect some other previously removed part. This would generate an iterative process which grows over time. Thus, some natural questions arise: what happens to this process over time? What conditions are relevant to the long-term behavior of this process? Can the outcome of this process ever be guaranteed? Let us assume that this membrane is infinite in size. Then, in purely mathematical terms, loss of tension corresponds to taking the convex hull of any intersecting parts of a membrane that have been removed, and this process would correspond to an iterative stochastic process which has some initial distribution of shapes within this membrane which are considered “removed” from the membrane. Then, at each step, the process removes the convex hull of overlapping removed sections; if this newly removed section overlaps another, the convex hull is taken again. Then, since this process is deterministic for any initial distribution of removed sections, the question becomes what can we say about this process over time based on the initial distribution of holes? When does this process continue forever, resulting in the entire plane being destroyed? And when does this process reach a state where no new parts get removed, resulting in part of the membrane remaining under tension? Is either outcome ever guaranteed? These questions are the central focus of this master’s thesis.