Anomalous First Passage in Evolution: Edge-KPZ Theory

The pace of evolution depends on how rapidly new phenotypes arise. We show that neutral Wright-Fisher evolution exhibits anomalous first passage despite diffusive mutations. The mean time for the first individual to reach a prescribed phenotypic distance scales approximately as $(σ^2)^{-3/2}$ with mutation variance $σ^2$. Two crossovers bound this regime, with inverse-variance scaling on either side. Combining coalescent theory with Kardar-Parisi-Zhang (KPZ) fluctuations at the dilute population edge, we develop an edge-KPZ theory of all three regimes. The anomaly persists under weak selection.