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arXiv · 2607.16489

Benjamini-Schramm limit of the heat semigroup on quantum graphs

Mihály Kovács·Eszter Sikolya
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Abstract

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.

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Mihály Kovács, Eszter Sikolya. 2026-07-17. Benjamini-Schramm limit of the heat semigroup on quantum graphs. https://arxiv.org/abs/2607.16489

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