The phase transition for domination in two interacting urns with power reinforcement

We study two interacting urns with power reinforcement $W(n)=n^α$, for $α>1$. Using Qin's stochastic-approximation theorem as input, we reduce the phase transition problem for domination to the sign analysis of a single analytic function $Φ_p$. This deterministic reduction identifies the sharp threshold and proves that the two natural critical parameters for domination coincide.