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ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS

Part of: Quantum field theory; related classical field theories Stochastic processes

Published online by Cambridge University Press:  09 October 2019

YAJUN ZHOU Open the ORCID record for YAJUN ZHOU [Opens in a new window]
YAJUN ZHOU*
Affiliation:
Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, NJ 08544, USA Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing 100871, PR China email yajunz@math.princeton.edu, yajun.zhou.1982@pku.edu.cn
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Abstract

Let $p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver’s probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-dimensional quantum field theory, we evaluate the third-order derivative $p_{5}^{\prime \prime \prime }(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $p_{n}(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $n\geq 5$, thus settling another conjecture of Borwein.

Keywords

Bessel functionsrandom walksWick rotationsTaylor expansions

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$
Secondary: 81T18: Feynman diagrams
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 3 , December 2019 , pp. 392 - 411
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).

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