Criticality: Subcritical, Critical, and Supercritical Regimes

The previous lesson established the single-number dichotomy at the heart of branching-process theory: extinction is certain if $\mu\le1$, and merely possible if $\mu>1$. This note names the three resulting regimes — subcritical, critical, supercritical — and looks at what distinguishes them beyond the single fact of certain-vs-possible extinction: how fast the subcritical process dies, why the critical case is a knife-edge, and how the supercritical process's growth is tamed into a genuine limiting random variable.

Learning objectives. By the end of this note you should be able to: