Zhao, YufeiGunby, Benjamin2021-07-1320212021-04-272021-05Gunby, Benjamin. 2021. Upper Tails of Subgraph Counts in Sparse Regular Graphs. Doctoral dissertation, Harvard University Graduate School of Arts and Sciences.28490755https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37368210What is the probability that a sparse $n$-vertex random $d$-regular graph $G_n^d$, $n^{1-c}=o(n)$ contains many more copies of a fixed graph $K$ than expected? We determine the behavior of this upper tail to within a logarithmic gap in the exponent. For most graphs $K$ (for instance, for any $K$ of average degree at least $4$) we determine the upper tail up to a $1+o(1)$ factor in the exponent. However, we also provide an example of a graph, given by adding an edge to $K_{2,4}$, where the upper tail probability behaves differently from previously studied behavior in both the sparse random regular and sparse Erd\H{o}s-R\'{e}nyi models in this sparsity regime.application/pdfenCombinatoricsGraph TheoryLarge DeviationsMathematicsThesis or Dissertation2021-07-130000-0002-3462-4618