RANDOM QUANTUM GRAPHS

We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini- Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples (X1, ,Xd) of traceless self-adjoint operators in the n × n matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: 2 ≤ d ≤ n2 - 3. Moreover, the automorphism group is generically abelian in the larger parameter range 1 ≤ d ≤ n2-2. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of Xi's (mimicking the Erdos-Rényi G(n, p) model) has trivial/abelian automorphism group almost surely.