Journal of Statistical Physics

, Volume 131, Issue 6, pp 1067–1083 | Cite as

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

  • Makoto Katori
  • Minami Izumi
  • Naoki Kobayashi


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.


Bessel process Bessel bridge Noncolliding diffusion process Riemann zeta function Jacobi theta function Double Dirichlet series Dyck path Vicious walk 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Physics, Faculty of Science and EngineeringChuo UniversityTokyoJapan

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