We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of DuanSeveriniWinter / Weaver) have few symmetries: for a Zariskidense open set of tuples ( X 1 , ⋯ , X d ) (X_1,\cdots ,X_d) of traceless selfadjoint operators in the n × n n\times n matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: 2 ≤ d ≤ n 2 − 3 2\le d\le n^23 . Moreover, the automorphism group is generically abelian in the larger parameter range 1 ≤ d ≤ n 2 − 2 1\le d\le n^22 . This then implies that for those respective parameters the corresponding randomquantumgraph model built on the GUE ensembles of X i X_i ’s (mimicking the ErdősRényi G ( n , p ) G(n,p) model) has trivial/abelian automorphism group almost surely.
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The free group on n generators modulo n + u random relations as n goes to infinity
Abstract We show that, as n goes to infinity, the free group on n generators, modulo {n+u} random relations, converges to a random group that we give explicitly. This random group is a nonabelian version of the random abelian groups that feature in the Cohen–Lenstra heuristics. For each n , these random groups belong to the few relator model in the Gromov model of random groups.
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 Award ID(s):
 2052036
 NSFPAR ID:
 10253279
 Date Published:
 Journal Name:
 Journal für die reine und angewandte Mathematik (Crelles Journal)
 Volume:
 2020
 Issue:
 762
 ISSN:
 00754102
 Page Range / eLocation ID:
 123 to 166
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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