Past Probability Seminars Spring 2020: Difference between revisions
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We study the distribution of the sandpile group of random d-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne. | We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne. | ||
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case. | Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case. | ||
Revision as of 20:42, 31 July 2018
Fall 2018
Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
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Friday, August 10, 10am, B239 Van Vleck András Mészáros, Central European University, Budapest
** Please note the unusual day, time, and room. **
Title: The distribution of sandpile groups of random regular graphs
Abstract: We study the distribution of the sandpile group of random [math]\displaystyle{ d }[/math]-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the [math]\displaystyle{ p }[/math]-Sylow subgroup of the sandpile group is a given [math]\displaystyle{ p }[/math]-group [math]\displaystyle{ P }[/math], is proportional to [math]\displaystyle{ |\operatorname{Aut}(P)|^{-1} }[/math]. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.
Our results extends a recent theorem of Huang saying that the adjacency matrices of random [math]\displaystyle{ d }[/math]-regular directed graphs are invertible with high probability to the undirected case.