Damage spreading in the random cluster model

TL;DR

This paper studies damage spreading in the random cluster model on 2D and 3D grids, identifying maxima in damage functions and exploring the chaotic phase boundary related to the model's parameters.

Contribution

It provides new insights into damage spreading behavior and phase boundaries in the random cluster model across different dimensions.

Findings

Damage function maxima identified at specific p-values for various q.

Chaotic phase boundary points are located in the (p,q) parameter space.

Lower bound of the chaotic phase may coincide with the Potts model's critical point for q≥3.

Abstract

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 $p=\sqrt{q}/(1+\sqrt{q})$ for all $q>0$ and also local maxima at $p=1/2$ and $p=q/(1+q)$ for $q\lesssim 0.75$. For 3D we observe a local maximum at $p=q/(1+q)$ for $q\lesssim 0.46$ and a global maximum at $p=1/2$ 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 $q\ge 3$.