论文标题
Benjamini-Schramm极限中量子图的经验光谱测量
Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
论文作者
论文摘要
我们介绍了量子图的本杰米尼·塞拉姆收敛的概念。这种融合的概念旨在扮演离散图的现有概念的角色，这意味着量子图限制对随机选择的球具有限制性分布。我们证明，具有均匀界限数据的任何量子图序列在这个意义上都具有收敛子序列。然后，我们考虑收敛序列的经验光谱度量（具有一般边界条件和边缘电势），并表明它会收敛到极限随机根量子图的预期光谱度量。这些结果与离散案例相似，但是证明明显不同。
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.
