Isoperimetric Properties of the Uniform Infinite Planar Triangulation
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We investigate some of the geometric properties of rooted uniform infinite planar triangulations or UIPT. We wish to establish certain isoperimetric properties of the UIPT - for example, to obtain some bounds on the boundary size of a connected subset of the UIPT containing the root. The results are contingent on some unproven results. We attempt to give some idea how these may be shown and why, in all likelihood, they are in fact true. Also, we will show a proof that if A is a simply connected subset of the plane consisting of a finite union of faces of the UIPT, then |∂A| ≥ cn¹/ⁿ for some constant c depending on ϵ, and where |A| ≥ n.
Duncan, Parker Alley (2018). Isoperimetric Properties of the Uniform Infinite Planar Triangulation. Master's thesis, Texas A & M University. Available electronically from