We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval I, in the sense that their spectrum in I is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in I are spatially delocalized.

中文翻译：

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在光谱离域范围内扩展量子图的量子遍历性

我们考虑具有很少循环的有限量子图序列，以便它们在 Benjamini-Schramm 的意义上收敛到随机无限量子树。我们假设这些量子树在某个区间I 中是光谱离域的，因为它们在I中的光谱纯粹是绝对连续的，并且它们的格林函数在实轴附近得到了很好的控制。我们进一步假设离散图的底层序列正在扩展。我们推导出量子遍历性结果，表明特征值位于I的特征函数在空间上是离域的。