CEU Electronic Theses and Dissertations, 2020
| Author | Simon, Peter Zoltan |
|---|---|
| Title | Central limit theorems for the winding and linking number |
| Summary | We aim to study the winding number of certain random curves and the linking number of pairs of random curves. Our motivation comes from a work of Liu, Dehmamy, and Barabasi, where the authors study the “tangledness” of graph embeddings. In order to test their hypothesis, they need to generate random embeddings of a given graph into R3 with many “self-linkings”. Once the image of each vertex is fixed, they replace each edge with a polygonal path (broken line) whose intermediate points are IID points chosen from a uniform distribution. Then they measure the tangledness of the embedding by considering the linking numbers of pairs of disjoint cycles of the graph. They make several empirical observations for these random embeddings. Our original goal was to study the distribution of the linking number of two random polygonal paths. Computer simulations suggested that, after proper normalization, it might converge to a normal distribution (as the number of IID intermediate points go to infinity). We started our investigations with a less complicated problem of similar flavor: the winding nuber of a random (closed) polygonal path on the plane. After expressing the winding number as the sum of a martingale difference sequence, we could rigorously prove that its distribution converges to a Gaussian by applying a Central Limit Theorem (CLT) for martingales. Then we turned to the linking number hoping to be able to prove a CLT for its distribution using similar tools. Now we believe that the limiting distribution is not quite normal. We do not have a rigorous proof at this point but our observations suggest that we may see an uncountable mixture of centered Gaussians in the limit. |
| Supervisor | Harangi, Viktor |
| Department | Mathematics MSc |
| Full text | https://www.etd.ceu.edu/2020/peter_simon02.pdf |
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