Moment bounds and exclusion processes on random Delaunay triangulations with conductances

TL;DR

This paper studies the properties of Delaunay triangulations derived from stationary point processes, focusing on conductance weights, integrability conditions, and implications for random walks and exclusion processes, including non-symmetric cases.

Contribution

It establishes integrability conditions for weighted degrees in Delaunay triangulations and extends the construction of exclusion processes to non-symmetric cases under certain conditions.

Findings

Integrability of weighted degrees and moments for Delaunay triangulations.

Conditions for constructing symmetric and non-symmetric exclusion processes.

Analysis of Bernoulli bond percolation on Delaunay triangulations.

Abstract

We consider the Voronoi tessellation associated to a stationary simple point process on $\mathbb{R}^d$ with finite and positive intensity. We introduce the Delaunay triangulation as its dual graph, i.e.~the graph with vertex set given by the point process and with edges between vertices whose Voronoi cells share a $(d-1)$-dimensional face. We also attach to each edge a random weight, called conductance. We provide sufficient conditions ensuring the integrability w.r.t. the Palm distribution of several quantities as weighted degrees and associated moments. These integrability properties are crucial in applications, as they allow to apply existing results on random walks, resistor networks and the symmetric simple exclusion processes with random conductances (cf. [1,11,12,13,14,15]). For the latter, while the moment bounds ensure its well definiteness and several properties, the same does not hold when the jump rates are not symmetric, i.e. for a generic simple exclusion process. In this case, by using a criterion from [13], we recover construction and properties of the simple exclusion process whenever the simple point process has finite range of dependence and the conductances are uniformly upper bounded. This last result relies on a suitable analysis of Bernoulli bond percolation on the Delaunay triangulation inspired by [2,3]. All the above results remain valid if the simple point process is stationary with respect to integer translations.