  The Voronoi construction is ubiquitous across the natural sciences and
engineering. In statistical mechanics, though, critical phenomena have so far
been only investigated on the Delaunay triangulation, the dual of a Voronoi
graph. In this paper we set to fill this gap by studying the two most prominent
systems of classical statistical mechanics, the equilibrium spin-1/2 Ising
model and the non-equilibrium contact process, on two-dimensional random
Voronoi graphs. Particular motivation comes from the fact that these graphs
have vertices of constant coordination number, making it possible to isolate
topological effects of quenched disorder from node-intrinsic coordination
number disorder. Using large-scale numerical simulations and
finite-size-scaling techniques, we are able to demonstrate that both systems
belong to their respective clean universality classes. Therefore, quenched
disorder introduced by the randomness of the lattice is irrelevant and does not
influence the character of the phase transitions. We report the critical points
of both models to considerable precision and, for the Ising model, also the
first correction-to-scaling exponent.
