  This paper is concerned with the limit laws of the extreme order statistics
derived from a symmetric Laplace walk. We provide two different descriptions of
the point process of the limiting extreme order statistics: a branching
representation and a squared Bessel representation. These complementary
descriptions expose various hidden symmetries in branching processes and
Brownian motion which lie behind some striking formulas found by Schehr and
Majumdar (Phys. Rev. Lett., 108:040601). In particular, the Bessel process of
dimension $4 = 2+2$ appears in the descriptions as a path decomposition of
Brownian motion at a local minimum and the Ray-Knight description of Brownian
local times near the minimum.
