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 $\alpha$) 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 $\alpha =0$ 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.

中文翻译：

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大等边量子图的量子遍历性

考虑在Benjamini–Schramm的意义上收敛到无限规则树的一系列有限规则图。我们研究了等边边长，基尔霍夫条件（可能具有非零耦合常数）的诱导量子图$\alpha$）和对称势 $ü$在边缘。我们证明在无限量子树具有绝对连续光谱的光谱区域中，会聚量子图的本征函数满足量子遍历定理。以防万一$\alpha =0$ 和 $ü=0$，极限度量是边缘上的统一度量。一般来说，它有一个明确的$C^{\mathrm{1个}}$密度。最后，我们证明了一个涉及积分算子的更强的量子遍历定理，其目的是研究本征函数相关性。