量子图上热半群的本雅明尼 - 施拉姆极限
Benjamini-Schramm limit of the heat semigroup on quantum graphs
AI总结：
研究量子图热半群在本雅明尼 - 施拉姆收敛下的行为，利用规范方法使热半群连续依赖有根量子图，通过相关逼近得到交换截断半径和图极限的双极限定理。
AI中文摘要：
我们研究了在本雅明尼 - 施拉姆收敛下量子图上热半群的行为。对于具有一致有界几何结构、配备连续性和基尔霍夫顶点条件的量子图，我们表明，通过边的规范广度优先识别传输到公共希尔伯特空间的热半群，相对于局部本雅明尼 - 施拉姆型度量，连续依赖于基础的有根量子图。因此，半群的根平均配对沿着有限量子图的本雅明尼 - 施拉姆收敛序列收敛。将此与通过度量球上的半群对该半群的 Trotter - Kato 型逼近相结合，我们得到了一个交换截断半径和图极限的双极限定理。
英文摘要：
We study the behaviour of heat semigroups on quantum graphs under Benjamini-Schramm convergence. For quantum graphs with uniformly bounded geometry, equipped with continuity and Kirchhoff vertex conditions, we show that the heat semigroup, transported to a common Hilbert space by a canonical breadth-first identification of the edges, depends continuously on the underlying rooted quantum graph with respect to a local Benjamini-Schramm-type metric. As a consequence, root-averaged pairings of the semigroup converge along Benjamini-Schramm convergent sequences of finite quantum graphs. Combining this with a Trotter-Kato-type approximation of the semigroup by semigroups on metric balls, we obtain a double-limit theorem interchanging the truncation radius and the graph limit.
