Short Walk Adventures

  • Armin Straub
  • Wadim ZudilinEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 313)


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.


Uniform random walk Mahler measure Modular function Modular form L-value Arithmetic differential equation Hypergeometric function 



We thank H. Cohen for supplying us with the numerical observation (10) whose origin remains completely mysterious to us. We also thank the referee for their valuable feedback.

Both authors acknowledge support of the Max Planck Society during their stays at the Max Planck Institute for Mathematics (Bonn, Germany) in the years 2015–2017. The second author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. no. 14.641.31.0001.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA
  2. 2.Department of Mathematics, IMAPPRadboud UniversityNijmegenNetherlands
  3. 3.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia
  4. 4.Laboratory of Mirror Symmetry and Automorphic FormsNational Research University Higher School of EconomicsMoscowRussia

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