Mathematics – Probability
Scientific paper
2007-11-12
J.Stat.Phys.131:1067-1083,2008
Mathematics
Probability
v2: LaTeX, 19 pages, 2 figures, minor corrections made for publication in J. Stat. Phys
Scientific paper
10.1007/s10955-008-9524-0
It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.
Izumi Minami
Katori Makoto
Kobayashi Naoki
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