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Distributional Lattices on Riemannian symmetric spaces. (arXiv:1707.00308v1 [math.PR])
来源于:arXiv
A Riemannian symmetric space is a Riemannian manifold in which it is possible
to reflect all geodesics through a point by an isometry of the space. On such
spaces, we introduce the notion of a distributional lattice, generalizing the
notion of lattice. Distributional lattices exist in any Riemannian symmetric
space: specifically the Voronoi tessellation of an stationary Poisson point
process is an example. We show that for an appropriate notion of amenability,
the amenability of a distributional lattice is equivalent to the amenability of
the ambient space. Using this equivalence, we show that the simple random walk
on any distributional lattice has positive embedded speed. For nonpositively
curved, simply connected spaces, we show that the simple random walk on a
Poisson--Voronoi tessellation has positive graph speed by developing some
additional structure for Poisson--Voronoi tessellations. 查看全文>>