Anchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaces

Abstract

We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation. arXiv:1409.4312v2 [math.PR]

First passage percolation on a hyperbolic graph admits bi-infinite geodesics

“…• (Poisson Voronoi and other random models) A variant of random metric perturbation is obtained via Poisson Voronoi tiling of a measure metric space. It seems likely that our method of proof applies to the hyperbolic Poisson Voronoi tiling, see [BPP14]. Recently other versions of random hyperbolic triangulations were constructed, [AR13] [C14].…”

Coalescing random walk on unimodular graphs

“…Augmented means that an extra child is added to the root. Note this is much more general than the exponential tail requirement needed in [ [BPP14]. Furthermore, combining our result with a simple compactness argument also yields that, whenever G is a finite graph whose degree distribution is stochastically dominated by some integrable reference distribution µ, CRW "looks recurrent" on G from the perspective of most vertices, in a quantitative way that depends only on the reference distribution µ.…”

Indistinguishability of trees in uniform spanning forests

“…We prove all of our results in the much more general setting of unimodular random rooted networks, which includes all Cayley graphs as well as a wide range of popular infinite random graphs and networks [2]. For example, our results hold when the underlying graph is an infinite supercritical percolation cluster in a Cayley graph, a hyperbolic unimodular random triangulation [10,5] (for which the FUSF and WUSF are shown to be distinct in the upcoming work [3]), a supercritical Galton-Watson tree, or even a component of the FUSF of another unimodular random rooted network. See Section 1.3 for the strongest and most general statements.…”

“…of Lemma 4.7 if e + , e − are both in T F (ρ) and part (1) otherwise, we deduce from the definition of δ-update-friendliness that for everyevent B ∈ {0, 1} E(G) such that WUSF G (F ∈ B) > 0, P (G,ρ) (E n e ∩ {F ∈ B}) = P (G,ρ) (E n e | F ∈ B)WUSF G (F ∈ B) ≥ δ P (G,ρ) (E n e | {U (F, e) ∈ B})WUSF G (F ∈ B) = δ1 c(e) c(e − ) ≥ δ WUSF G (F ∈ B) WUSF G (U (F, e) ∈ B) · P (G,ρ) (E n e ∩ {U (F, e) ∈ B}) ≥ δ 2 P (G,ρ) (E n e ∩ {U (F, e) ∈ B}) ,(4 5). …”