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On certain random topological structures

Vander Werf, Andrew Michael
http://orcid.org/0009-0008-9920-4463
http://rave.ohiolink.edu/etdc/view?acc_num=osu168192057534709

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
This thesis explores the properties of two particular random topological structures: the random geometric graph on high-dimensional spheres and k-dimensional hypertrees distributed according to their natural determinantal probability measure. The contents of two papers are included, each discussing one of these structures. Our results on the random geometric graph on n vertices drawn uniformly from a d-dimensional sphere concern its containment probabilities, that is the probability that this graph contains a given deterministic subgraph G. We show for a substantial family of graphs G that as d tends to infinity this statistic is asymptotically equivalent to that of the Erdős-Rényi random graph of (roughly) the same edge density. To do the analysis, we derive some exact statistics of the spherical Wishart matrix, the Gram matrix of n independent uniformly random d-dimensional spherical vectors. In particular we give expressions for the characteristic function of the spherical Wishart matrix which are well-approximated using steepest descent. In our exploration of k-dimensional hypertrees, we deduce a structurally inductive description of the determinantal probability measure associated with Kalai's celebrated enumeration result for higher-dimensional spanning trees of the n-1-simplex. As a consequence, we derive the marginal distributions of the simplex links in such random trees. Along the way, we also characterize the higher-dimensional spanning trees of every other simplicial cone in terms of the higher-dimensional rooted forests of the underlying simplicial complex. We also apply these new results to random topology, the spectral analysis of random graphs, and the theory of high-dimensional expanders. One particularly interesting corollary of these results is that the fundamental group of a union of o(logn) determinantal 2-trees has Kazhdan's property (T) with high probability.
Matthew Kahle (Advisor)
101 p.
Mathematics

Recommended Citations

Citations

  • Vander Werf, A. M. (2023). On certain random topological structures [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu168192057534709

    APA Style (7th edition)

  • Vander Werf, Andrew. On certain random topological structures. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu168192057534709.

    MLA Style (8th edition)

  • Vander Werf, Andrew. "On certain random topological structures." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu168192057534709

    Chicago Manual of Style (17th edition)

osu168192057534709
45
© 2023, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.