## Abstract

Consider a sequence of finite regular graphs converging, in the sense of Benjamini–Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant α) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case α = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it has an explicit C^{1} density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations.

Original language | English (US) |
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Pages (from-to) | 82-109 |

Number of pages | 28 |

Journal | Journal of the London Mathematical Society |

Volume | 101 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2020 |

## Keywords

- 34B45
- 58J51 (primary)
- 81Q10 (secondary)

## ASJC Scopus subject areas

- General Mathematics