量子图在Benjamini-Schramm极限下的经验谱测度
Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
摘要：
我们引入量子图的Benjamini-Schramm收敛的概念。这个收敛的概念，旨在扮演已经存在的离散图概念的角色，意味着量子图对随机选择的球的限制具有限制分布。我们证明了任何序列的量子图一致有界的数据有一个收敛的子序列在这个意义上。然后，我们考虑经验谱测度的收敛序列（一般边界条件和边缘电位），并表明它收敛到预期的谱测度的限制随机根量子图。这些结果类似于离散的情况，但证明是显着不同的。
英文摘要：
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.
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