There are several functional encodings of random trees which are commonly used to prove (among other things) scaling limit results. We consider two of these, the height process and Lukasiewicz path, in the classical setting of a branching process tree with critical offspring distribution of finite variance, conditioned to have n vertices. These processes converge jointly in distribution after rescaling by n^{-1/2} to constant multiples of the same standard Brownian excursion, as n goes to infinity. Their difference (taken with the appropriate constants), however, is a nice example of a discrete snake whose displacements are deterministic given the vertex degrees; to quote Marckert, it may be thought of as a “measure of internal complexity of the tree”. We prove that this discrete snake converges on rescaling by n^{-1/4} to the Brownian snake driven by a Brownian excursion. We believe that our methods should also extend to prove convergence of a broad family of other “globally centred” discrete snakes which seem not to be susceptible to the methods of proof employed in earlier works of Marckert and Janson.
This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.