= Product decay in a cyclic imitation chain
{title2=$\mathbb E P(X_t^N)=e^{-3t/N}\mathbb E P(X_0^N)$}
For the <cyclic imitation chain>, its <infinitesimal generator> satisfies $\mathcal L_N(x_1x_2x_3)=-3x_1x_2x_3/N$. Therefore $\mathbb E[X_1^N(t)X_2^N(t)X_3^N(t)]=e^{-3t/N}\mathbb E[X_1^N(0)X_2^N(0)X_3^N(0)]$. While all three types remain, the <product of numbers> is at least $(N-2)/N^3$ for $N\geq3$, so this identity proves that one type is lost almost surely. The remaining pair has a fixed winner, which eventually occupies the whole population. The conserved product in the deterministic <fluid limit> therefore describes fixed finite horizons, not the chain's infinite-time behavior.
Back to article page
