Conjecture on the asymptotic distribution of the longest expanded cycle

Let

be the spatial period and $r$ the neighborhood radius in the random-rule cellular automaton, and let $C_n$ denote the longest expanded cycle of its de Bruijn-like expanded cycle graph after cemetery vertices are eliminated. Fix arbitrary $,r\text{?}$ Wait. Let

be the spatial period parameter and $r\geq 1$ the neighborhood radius, and consider the limit as $n\to\infty$. The asymptotic cycle-length conjecture. If

$$is odd, then$$

n^{-}C_n\to 1

$$in probability. If$$

is even, then $n^{-}C_n$ converges in distribution to a nontrivial bimodal distribution. The conjecture extends the results proved when the spatial period does not exceed the number of neighbors; simulations suggest that the same parity-dependent behavior persists when $>r$, where dependencies among arcs make a rigorous analysis difficult.