Abstract
We prove that the extremal process of branching Brownian motion, in the limit
of large times, converges weakly to a cluster point process. The limiting
process is a (randomly shifted) Poisson cluster process, where the positions of
the clusters is a Poisson process with exponential density. The law of the
individual clusters is characterized as branching Brownian motions conditioned
to perform "unusually large displacements", and its existence is proved. The
proof combines three main ingredients. First, the results of Bramson on the
convergence of solutions of the Kolmogorov-Petrovsky-Piscounov equation with
general initial conditions to standing waves. Second, the integral
representations of such waves as first obtained by Lalley and Sellke in the
case of Heaviside initial conditions. Third, a proper identification of the
tail of the extremal process with an auxiliary process, which fully captures
the large time asymptotics of the extremal process. The analysis through the
auxiliary process is a rigorous formulation of the cavity method developed in
the study of mean field spin glasses.
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