The Brownian snake and random trees

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.