Convergence towards Ideal Poisson--Voronoi tessellations with a focus on Diestel--Leader graphs

We provide necessary and sufficient conditions for convergence towards a unique IPVT on any proper pointed measured metric space.

The conditions are that the volume function, when composed with $\log$, is regularly varying and that the limit of the uniform probability measure on a large ball exists in the horocompactification.

As an application we prove convergence towards a unique IPVT for higher rank symmetric spaces, which solves an open problem of \cite{MiMe23}.

Versions of this theorem are provided for graphs and edge-measured graphs, where a natural parameter $\xi$ appears.

We prove independence on $\xi$ in a specific sense under mild assumptions, which answers an open problem of~\cite{IPVT}.

As a main example, we show that the latter holds for the IPVT of Diestel-Leader graphs.

We also focus on further properties of this example, in particular, that its IPVT cells are distinguishable, providing the first Cayley graph with this property.