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Homological Percolation in a Torus

Duncan, Paul
http://orcid.org/0000-0002-2663-7275
http://rave.ohiolink.edu/etdc/view?acc_num=osu1655122886253287

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis we study extensions to higher dimensions of several versions of percolation within a torus. We include the contents of two papers, one covering independent percolation and the other covering a well-known dependent model. Percolation traditionally studies the appearance of infinite components in random subgraphs of lattices. The canonical subgraphs studied this way are obtained by taking a constant fraction of the vertices or edges independently at random. One can build higher dimensional random complexes in this way, but it is not clear what the equivalent of the infinite component should be. Bobrowski and Skraba offered a partial answer in finite volume complexes called homological percolation. In the torus T^d, homological percolation in dimension i means that the subcomplex contains a representative of a nontrivial element of H_i(T^d). In section 2 we consider analogues of both bond and site percolation in the torus and show that such percolation has a sharp threshold function in all dimensions. We also show that percolation in half the dimension of the torus occurs at p=1/2, analogous to the classical Harris-Kesten theorem. Another percolation model of interest is the random-cluster model, which weights configurations of independent percolation according to the number of connected components. This is a particularly interesting model because it can be coupled with the Ising and Potts models of magnetism. In section 3 we study a higher dimensional version of this model introduced by Hiraoka and Shirai, which also admits a coupling to a higher dimensional Potts model. It is worth noting that the two dimensional random-cluster model is associated to a Potts lattice gauge theory, which is related to interesting questions from physics. We prove similar results about sharp thresholds for homological percolation in the random-cluster model, which sheds some light on the Wilson loops in the associated lattice gauge theory.
Matthew Kahle (Advisor)
David Sivakoff (Committee Member)
Facundo Mémoli (Committee Member)
120 p.
Mathematics
percolation; plaquette percolation; homological percolation; random cluster model; Potts model; lattice gauge theory

Recommended Citations

Citations

  • Duncan, P. (2022). Homological Percolation in a Torus [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1655122886253287

    APA Style (7th edition)

  • Duncan, Paul. Homological Percolation in a Torus. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1655122886253287.

    MLA Style (8th edition)

  • Duncan, Paul. "Homological Percolation in a Torus." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1655122886253287

    Chicago Manual of Style (17th edition)

osu1655122886253287
135
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This open access ETD is published by The Ohio State University and OhioLINK.