The Brownian snake (with lifetime given by a normalized
Brownian excursion) arises as a natural limit when studying random trees. This
may be used in both directions, i.e. to obtain asymptotic results for random
trees in terms of the Brownian snake, or, conversely, to deduce properties of
the Brownian snake from asymptotic properties of random trees. The arguments
are based on Aldous' theory of the continuum random tree.
I will discuss two such situations:
1. The Wiener index of random trees converges, after
suitable scaling, to the integral (=mean position) of the head of the Brownian
snake. This enables us to calculate the moments of this integral.
2. A branching random walk on a random tree converges, after
suitable scaling, to the Brownian snake, provided the distribution of the
increments does not have too large tails. For i.i.d increments Y with mean 0,
a necessary and sufficient condition is that the tails are o(y^{-4}); in
particular, a finite fourth moment is enough, but weaker moment conditions are
not.