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 isFurther, it is proved independently in Adkon 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 2t} with < 1, which reveals a phase transition at = 1 - 2. We also verify the conjecture of Derrida and Shi (2017b) on the precise asymptotic behaviour of P(M t 2t) for < 1.