Random quantum graphs

TL;DR

This paper demonstrates that generic quantum graphs, constructed from random traceless self-adjoint operators, typically have trivial or abelian automorphism groups, indicating low symmetry in these models.

Contribution

It establishes that for a broad class of quantum graphs, the automorphism group is almost surely trivial or abelian, extending understanding of symmetry properties in quantum graph models.

Findings

Most quantum graphs have trivial automorphism groups.

Automorphism groups are generically abelian in larger parameter ranges.

Random quantum graphs built from GUE ensembles have low symmetry almost surely.

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 Erd\H{o}s-R\'{e}nyi $G(n,p)$ model) has trivial/abelian automorphism group almost surely.