Quenched invariance principle for random walks on Delaunay triangulations

We consider simple random walks on Delaunay triangulations generated by point processes in ${\mathbb{R}}^d$. Under suitable assumptions on the point processes, we show that the random walk satisfies an almost sure (or quenched) invariance principle. This invariance principle holds for point processes which have clustering or repulsiveness properties including Poisson point processes, Mat{'e}rn cluster and Mat{'e}rn hardcore processes. The method relies on the decomposition of the process into a martingale part and a corrector which is proved to be negligible at the diffusive scale.