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

We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of degree d, depending on repulsion strength beta between particles of di?fferent signs and on an activity parameter lambda 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 beta. 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 greater or equal 2. In our second and main result, we also show that for large beta and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity increases, assuming that d is greater or equal than 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.