We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a randomly chosen ball has a limiting distribution. We prove that any sequence of quantum graphs with uniformly bounded data has a convergent subsequence in this sense. We then consider the empirical spectral measure of a convergent sequence (with general boundary conditions and edge potentials) and show that it converges to the expected spectral measure of the limiting random rooted quantum graph. These results are similar to the discrete case, but the proofs are significantly different.

我们介绍了量子图的本杰米尼-施拉姆收敛概念。这种趋同的概念旨在发挥离散图已经存在的概念的作用，这意味着将量子图限制在随机选择的球上具有极限分布。我们证明，从这个意义上讲，具有均匀边界数据的任何量子图序列都具有收敛的子序列。然后，我们考虑会聚序​​列（具有一般边界条件和边缘电势）的经验光谱测度，并表明它收敛于有限随机根量子图的预期光谱测度。这些结果与离散情况相似，但证明有很大不同。