Phase transition in LAWS convergence

Let $P_{\mathcal{M}}$ be a stationary distribution with entropy $H$, let $H_N$ denote the LAWS hit rate after $N$ total queries, and let $N_{\min}$ be the per-node visit threshold. Phase transition in LAWS convergence. There exists $N^*=\Theta(N_{\min}\cdot2^H)$ such that $H_N\approx0$ for $N\ll N^*$ and $H_N\approx H_\infty$ for $N\gg N^*$, with transition width $O(N^*/\sqrt{H})$.

The heuristic compares the process with coupon collection over $2^H$ heavy nodes, while noting that trie dependencies may accelerate coverage. A formal proof via the trie heavy-node occupancy process remains open.