Damage spreading in the random cluster model

We investigate the damage spreading effect in the Fortuin-Kasteleyn random cluster model for 2- and 3-dimensional grids with periodic boundary. For 2D the damage function has a global maximum at $p=\sqrt{q}\mathrm{/}(1+\sqrt{q})$ for all $q>0$ and also local maxima at $p=1\mathrm{/}2$ and $p=q\mathrm{/}(1+q)$ for $q\lesssim 0.75$. For 3D we observe a local maximum at $p=q\mathrm{/}(1+q)$ for $q\lesssim 0.46$ and a global maximum at $p=1\mathrm{/}2$ for $q\lesssim 4.5$. The chaotic phase of the model’s $(p,q)$-parameter space is where the coupling time is of exponential order and we locate points on its boundary. For 3-dimensional grids the lower bound of this phase may be equal to the corresponding critical point of the $q$-state Potts model for $q\ge 3$.