Dynamical Gibbs–non-Gibbs transitions in Widom–Rowlinson models on trees

Abstract

We consider the soft-core Widom–Rowlinson model for particles with spins and holes, on a Cayley tree of order d (which has d + 1 nearest neighbours), depending on repulsion strength β between particles of different signs and on an activity parameter λ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength β. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any d ≥ 2. In our second and main result, we also show that for large β and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity λ increases, assuming that d ≥ 4. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.