Asymptotic behaviour of watermelons

A watermelon is a set of $p$ Bernoulli paths starting and ending at the same ordinate, that do not intersect. In this paper, we show the convergence in distribution of two sorts of watermelons (with or without wall condition) to processes which generalize the Brownian bridge and the Brownian excursion in ${\mathbb{R}}^p$. These limit processes are defined by stochastic differential equations. The distributions involved are those of eigenvalues of some Hermitian random matrices. We give also some properties of these limit processes.