Stars and Watermelons

Gillet [3] demonstrated that limn→∞Wk(b2n tc)/ √ 2n tends to a family of p nonintersecting Brownian excursions, 0 ≤ t ≤ 1, as an extension of a principle given in [4]. The height of a path Wk in a p-watermelon with wall is the maximum value of Wk(i) over all i. The area under a path Wk is the area of the polygonal region determined by the curve j = Wk(i), the horizontal line j = 0, and the vertical lines i = 0, i = 2n. In the case p = 2, we will refer to the upper height and upper area (corresponding to W2) and the lower height and lower area (corresponding to W1). Counting all 1-watermelons with wall (or Dyck paths) and 2-watermelons with wall give (2n)! n!(n+ 1)! , 6(2n)!(2n+ 2)! n!(n+ 1)!(n+ 2)!(n+ 3)! possible configurations of length 2n, respectively. (The former is the n Catalan number.) The average height H1(n) for 1-watermelons with wall satisfies [5, 6] H1(n) ∼ √ πn