Densities of Short Uniform Random Walks

Borwein, Jonathan M.; Straub, Armin; Wan, J. G.; Zudilin, Wadim; Zagier, Don

doi:10.4153/cjm-2011-079-2

Cited by 59 publications

(122 citation statements)

References 21 publications

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“…This is illustrated in Figure 2 on the real line and later in Figure 3 on the complex plane. Notice that both W 3 (s) and W 4 (s) do not have zeroes for negative odd integers; however, as shown in [15], W 3 (−2k − 1) decays to 0 very rapidly as k → ∞.…”

Section: General Moments Of Random Walksmentioning

confidence: 99%

“…However, it should be noted that care [15] is needed when rigorously establishing that the densities p n satisfy the formally obtained differential equations: for instance, p 4 (x) is approximated by a constant multiple of √ 4 − x as x → 4 − , so that the second derivative of p 4 is not even locally integrable. In particular, p 4 does not have a Mellin transform in the classical sense.…”

Section: Densities Of Random Walksmentioning

confidence: 99%

“…In general, it is shown in [15] that the density p n satisfies a Fuchsian differential equation of order n − 1. From the leading coefficient of this differential equation one finds that p n is real analytic except at 0 and the positive integers n, n − 2, n − 4, .…”

Section: Densities Of Random Walksmentioning

confidence: 99%

“…In the case n = 3 a more tractable hypergeometric representation can be given. Starting with a Bessel integral representation valid for even moments and applying Carlson's Theorem, it is deduced in [15] that, for s not a negative integer < −1,…”

Section: Evaluations Of the First Moments For N = 3mentioning

confidence: 99%

“…These moments have been studied in [11,14], while the (radial) densities p n of the distance traveled in n steps have been the subject of [15]. The main results will be presented in Section 2.…”

Section: Introductionmentioning

confidence: 99%

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Mahler measures, short walks and log-sine integrals

Borwein

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

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The Mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years -very few closed forms are proven but more are conjectured. In the case of multiple Mahler measures more tractable but interesting families exist. Using values of log-sine integrals we provide systematic evaluations of various higher and multiple Mahler measures.The evaluations in terms of log-sine integrals become particularly useful in light of the fact that log-sine integrals may be automatically reexpressed as polylogarithmic values. We present this correspondence along with related generating functions for log-sine integrals.Our initial interest in considering Mahler measures stems from a study of uniform random walks in the plane as first introduced by Pearson. The main results on the moments of the distance traveled by an n-step walk, as well as the corresponding probability density functions, are reviewed. It is the derivative values of the moments that are Mahler measures.This work would be impossible without very extensive symbolic and numeric computations. It also makes frequent use of the new NIST Handbook of Mathematical Functions and similar tools. Our intention is to show off the interplay between numeric and symbolic computing while exploring the three mathematical topics in the title.

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Section: General Moments Of Random Walksmentioning

confidence: 99%

Section: Densities Of Random Walksmentioning

confidence: 99%

Section: Densities Of Random Walksmentioning

confidence: 99%

Section: Evaluations Of the First Moments For N = 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mahler measures, short walks and log-sine integrals

Borwein

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

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Some Algebraic and Arithmetic Properties of Feynman Diagrams

Zhou¹

2019

Texts &Amp; Monographs in Symbolic Computation

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This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagrams and critical values of modular L-functions.

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Short Walk Adventures

Straub

Zudilin

2020

Springer Proceedings in Mathematics &Amp; Statistics

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We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend available "probabilistic" techniques to cover a variation of random walks and reduce some three-variable Mahler measures, which are conjectured to evaluate in terms of L-values of modular forms, to hypergeometric form.

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Densities of Short Uniform Random Walks

Cited by 59 publications

References 21 publications

Mahler measures, short walks and log-sine integrals

Mahler measures, short walks and log-sine integrals

Some Algebraic and Arithmetic Properties of Feynman Diagrams

Short Walk Adventures

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