Branching Brownian motion conditioned on small maximum

For a standard binary branching Brownian motion on the real line, it is known that the typical value of the maximal position M-t among all particles alive at time t is m(t) + Theta(1) with m(t) = root 2t - 3/2 root 2 log t. Further, it is proved independently in Aidekon et al. (2013) and Arguin et al. (2013) that the branching Brownian motion shifted by m(t) (or M-t) converges in law to some decorated Poisson point process. The goal of this work is to study the branching Brownian motion conditioned on M-t << m(t). We give a complete description of the limiting extremal process conditioned on {M-t <= root 2 alpha t} with alpha < 1, which reveals a phase transition at alpha = 1 - root 2. We also verify the conjecture of Derrida and Shi (2017b) on the precise asymptotic behaviour of P(M-t <= root 2 alpha t) for alpha < 1.