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

Abstract

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.

v2: LaTeX, 19 pages, 2 figures, minor corrections made for publication in J. Stat. Phys

References in corpus (11)

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Random Matrix Theory of a Chaotic Andreev Quantum Dot
• Vicious walkers, friendly walkers and Young tableaux II: With a wall
• Scaling limit of vicious walks and two-matrix model
• Nonintersecting Brownian excursions
• Noncolliding Brownian Motion and Determinantal Processes
• Functional central limit theorems for vicious walkers
• Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals
• Watermelon configurations with wall interaction: exact and asymptotic results
• Asymptotic behaviour of watermelons

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