Particles diffuse and branch (split) at rate β, creating an exponentially growing population spreading as a traveling wave. The front velocity is v = √(2Dβ) — the KPP/Fisher traveling wave connects to this extremal statistics problem.
McKean (1975): rightmost particle in BBM converges to KPP traveling wave. Bramson (1978): correction m(t) = √(2β)t − (3/2√(2β))ln(t) + O(1). Connection to extremal random matrices and log-correlated fields.