Quenched invariance principle for random walks on Poisson-Delaunay triangulations

Abstract:

The Voronoi tiling of an infinite locally finite subset $\xi$ of ${\Bbb R}^d$ is the collection of the Voronoi cells:
$$
Vor_{\xi}(x)=( y \in \mathbb{R}^d : |y-x| \leq |y-x' |, \forall x' \in \xi ), \quad \quad x \in \xi.
$$
The associated Delaunay triangulation $\operatorname{DT}(\xi)$ is its dual graph in which there is an edge between vertices $x$ and $x'$ if $Vor_\xi(x)$ and $Vor_\xi(x)$ share a $(d-1)$-dimensional face.
When $\xi$ is distributed according to a Poisson point process, this graph is called Poisson-Delaunay triangulation.

In this talk, we present a quenched invariance principle for the variable speed nearest-neighbor random walk $(X_t)$ on the Delaunay triangulation of a realization $\xi$ of a Poisson point process, that is for the Markov process with generator:
$$
L^\xi f(x):=\sum_{y\in\xi} {\bf 1}_{y\sim x\text{ in }DT(\xi)}\left(f(y)-f(x)\right),\quad \quad x\in\xi.
$$

In other words, we show that, for almost every realization $\xi$ of the point process and all starting point $x\in\xi$, the rescaled process
$$
(X^\varepsilon_t){t\geq 0}=(\varepsilon X{\varepsilon^{-2}t})_{t\geq 0}
$$
converges in law as $\varepsilon$ tends to $0$ to a non degenerate Brownian motion under the quenched law.