Fine asymptotics for the consistent maximal displacement of branching Brownian motion

It is well-known that the maximal particle in a branching Brownian motion sits near $\sqrt{2}t-\frac{3}{2\sqrt{2}}\log t$ at time $t$. One may then ask about the paths of particles near the frontier: how close can they stay to this critical curve? Two different approaches to this question have been developed. We improve upon the best-known bounds in each case, revealing new qualitative features including marked differences between the two approaches.