Random quantum graphs

Chirvasitu, Alexandru; Wasilewski, Mateusz

doi:10.1090/tran/8584

Random quantum graphs
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by Alexandru Chirvasitu and Mateusz Wasilewski PDF

Trans. Amer. Math. Soc. 375 (2022), 3061-3087 Request permission

Abstract:

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 $(X_1,\cdots ,X_d)$ of traceless self-adjoint operators in the $n\times n$ matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: $2\le d\le n^2-3$. Moreover, the automorphism group is generically abelian in the larger parameter range $1\le d\le n^2-2$. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of $X_i$’s (mimicking the Erdős-Rényi $G(n,p)$ model) has trivial/abelian automorphism group almost surely.

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