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 order $d$ (which has $d + 1$ nearest neighbours), depending repulsion strength $\beta$ between different signs an activity parameter $\lambda$ particles. analyse Gibbsian properties time-evolved intermediate Gibbs measure static model, under spin-flip time evolution, in regime large $\beta$. first show that there is dynamical transition, which becomes non-Gibbsian at times, independently particle activity, any \geq 2$. In our second main result, we also set bad configurations (discontinuity points) changes from zero to one as increases, assuming 4$. Our proof relies general zero-one law tree, introduction uniformly given terms subtree percolation, become typical high activity.