  We consider the multicomponent Widom-Rowlison with Metropolis dynamics, which
describes the evolution of a particle system where $M$ different types of
particles interact subject to certain hard-core constraints. Focusing on the
scenario where the spatial structure is modeled by finite square lattices, we
study the asymptotic behavior of this interacting particle system in the
low-temperature regime, analyzing the tunneling times between its $M$
maximum-occupancy configurations, and the mixing time of the corresponding
Markov chain. In particular, we develop a novel combinatorial method that,
exploiting geometrical properties of the Widom-Rowlinson configurations on
finite square lattices, leads to the identification of the timescale at which
transitions between maximum-occupancy configurations occur and shows how this
depends on the chosen boundary conditions and the square lattice dimensions.
