Phase transition in one-dimensional excitable media with variable interaction range

We investigate two discrete models of excitable media on a one-dimensional integer lattice $\mathbb{Z}$: the $\kappa$-color Cyclic Cellular Automaton (CCA) and the $\kappa$-color Firefly Cellular Automaton (FCA). In both models, sites are assigned uniformly random colors from $\mathbb{Z}/\kappa\mathbb{Z}$. Neighboring sites with colors within a specified interaction range $r$ tend to synchronize their colors upon a particular local event of 'excitation'. We establish that there are three phases of CCA/FCA on $\mathbb{Z}$ as we vary the interaction range $r$. First, if $r$ is too small (undercoupled), there are too many non-interacting pairs of colors, and the whole graph $\mathbb{Z}$ will be partitioned into non-interacting intervals of sites with no excitation within each interval. If $r$ is within a sweet spot (critical), then we show the system clusters into ever-growing monochromatic intervals. For the critical interaction range $r=\lfloor \kappa/2 \rfloor$, we show the density of edges of differing colors at time $t$ is $\Theta(t^{-1/2})$ and each site excites $\Theta(t^{1/2})$ times up to time $t$. Lastly, if $r$ is too large (overcoupled), then neighboring sites can excite each other and such 'defects' will generate waves of excitation at a constant rate so that each site will get excited at least at a linear rate. For the special case of FCA with $r=\lfloor 2/\kappa \rfloor+1$, we show that every site will become $(\kappa+1)$-periodic eventually.