Skip to main content Accessibility help
×
Home

Densities of Short Uniform Random Walks

  • Jonathan M. Borwein (a1), Armin Straub (a2), James Wan (a1), Wadim Zudilin (a1) and Don Zagier (a3)...

Abstract

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and, less completely, those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Densities of Short Uniform Random Walks
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Densities of Short Uniform Random Walks
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Densities of Short Uniform Random Walks
      Available formats
      ×

Copyright

References

Hide All
[Aka09] Akatsuka, H., Zeta Mahler measures. J. Number Theory 129(2009), no. 11, 27132734. http://dx.doi.org/10.1016/j.jnt.2009.05.007
[BB10] Bailey, D. H. and Borwein, J. M., Hand-to-hand combat with thousand-digit integrals. J. Computational Science, 2010, http://dx.doi.org/10.1016/j.jocs.2010.12.004
[BBBG08] Bailey, D. H., Borwein, J. M., Broadhurst, D. J., and Glasser, M. L., Elliptic integral evaluations of Bessel moments and applications. J. Phys. A 41(2008), no. 20, 52035231.
[BB91] Borwein, J. M. and Borwein, P. B., A cubic counterpart of Jacobi's identity and the AGM. Trans. Amer. Math. Soc. 323(1991), no. 2, 691701. http://dx.doi.org/10.2307/2001551
[BB98] Borwein, J. M. and Borwein, P. B., Pi and the AGM. A study in analytic number theory and computational complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts, 4, John Wiley & Sons, Inc., New York, 1998.
[BBG94] Borwein, J. M., Borwein, P. B., and Garvan, F., Some cubic modular identities of Ramanujan. Trans. Amer. Math. Soc. 343(1994), no. 1, 3547. http://dx.doi.org/10.2307/2154520
[BNSW11] Borwein, J. M., Nuyens, D., Straub, A., and Wan, J., Some arithmetic properties of random walk integrals. Ramanujan J. 26(2011), no. 1, 109132. http://dx.doi.org/10.1007/s11139-011-9325-y
[Boy81] Boyd, D. W., Speculations concerning the range of Mahler's measure. Canad. Math. Bull. 24(1981), no. 4, 453469. http://dx.doi.org/10.4153/CMB-1981-069-5
[BS11] Borwein, J. M. and Straub, A., Log-sine evaluations of Mahler measures. J. Aust Math. Soc., to appear. arxiv:1103.3893.
[BSW11] Borwein, J. M., Straub, A., and Wan, J., Three-step and four-step random walk integrals. Experimental Mathematics, 2011, to appear. http://www.carma.newcastle.edu.au/_jb616/walks2.pdf.
[BZ92] Borwein, J. M. and Zucker, I. J., Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12(1992), no. 4, 519526. http://dx.doi.org/10.1093/imanum/12.4.519
[BZB08] Borwein, J. M., Zucker, I. J., and Boersma, J., The evaluation of character Euler double sums. Ramanujan J. 15(2008), no. 3, 377405. http://dx.doi.org/10.1007/s11139-007-9083-z
[CZ10] Chan, H. H. and Zudilin, W., New representations for Apéry-like sequences. Mathematika 56(2010), 107117. http://dx.doi.org/10.1112/S0025579309000436
[Cra09] Crandall, R. E., Analytic representations for circle-jump moments. PSIpress, 21 nov 09: E, http://www.perfscipress.com/papers/Analytic Wn psipress.pdf.
[DM04] Djakov, P. and Mityagin, B., Asymptotics of instability zones of Hill operators with a two term potential. C. R. Math. Acad. Sci. Paris 339(2004), no. 5, 351354. http://dx.doi.org/10.1016/j.crma.2004.06.019
[DM07] Djakov, P. and Mityagin, B., Asymptotics of instability zones of the Hill operator with a two term potential. J. Funct. Anal. 242(2007), no. 1, 157194. http://dx.doi.org/10.1016/j.jfa.2006.06.013
[Fet63] Fettis, H. E., On a conjecture of Karl Pearson. In: Rider anniversary volume, Defense Technical Information Center, Belvoir, VA, 1963, pp. 3954.
[Fin05] Finch, S., Modular forms on SL2(Z). http://algo.inria.fr/csolve/frs.pdf
[FS09] Flajolet, P. and Sedgewick, R., Analytic combinatorics. Cambridge University Press, Cambridge, 2009.
[Hör90] Hörmander, L., The analysis of linear partial differential operators. I. Second ed. Springer, Berlin, 1990.
[Hug95] Hughes, B. D., Random walks and random environments. Vol. 1. Random walks. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.
[Inc26] Ince, E. L., Ordinary differential equations. Green and Co., London, 1926.
[Klu05] Kluyver, J. C., A local probability problem. In: Royal Netherlands Academy of Arts and Sciences, Proceedings, 8 I, 1905, pp. 341350.
[Luk69] Luke, Y. L., The special functions and their approximations. Vol. 1, Academic Press, New York-London, 1969.
[ML86] Misra, O. P. and Lavoine, J. L., Transform analysis of generalized functions. North-Holland Mathematical Studies, 119, North-Holland Publishing Co., Amsterdam, 1986.
[Nes03] Nesterenko, Y. V.. Integral identities and constructions of approximations to zeta-values. J. Théor. Nombres Bordeaux 15(2003), no. 2, 535550. http://dx.doi.org/10.5802/jtnb.412
[Pea05a] Pearson, K., The problem of the random walk. Nature 72(1905), no. 1867, 342.
[Pea05b] Pearson, K., The problem of the random walk. Nature 72(1905), no. 1865, 294.
[Pea06] Pearson, K., A mathematical theory of random migration. In: Drapers company research memoirs, Biometric Series, III, Cambridge University Press, Cambridge, 1906.
[PWZ96] Petkovsek, M., Wilf, H., and Zeilberger, D., A = B. Peters, A. K., Wellesley, MA, 1996.
[Ray05] Rayleigh, The problem of the random walk. Nature 72(1905), no. 1866, 318.
[Rog09] Rogers, M. D., New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/. Ramanujan J. 18(2009), no. 3, 327340. http://dx.doi.org/10.1007/s11139-007-9040-x
[RVTV04] Rodriguez-Villegas, F., Toledano, R., and Vaaler, J. D., Estimates for Mahler's measure of a linear form. Proc. Edinb. Math. Soc. 47(2004), no. 2, 473494. http://dx.doi.org/10.1017/S0013091503000701
[SC67] Selberg, A. and Chowla, S., On Epstein's zeta-function. J. Reine Angew. Math. 227(1967), 86110. http://dx.doi.org/10.1515/crll.1967.227.86
[Tit39] Titchmarsh, E. C., The theory of functions. Second ed., Oxford University Press, 1939.
[Ver04] Verrill, H. A., Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations. arxiv:math/0407327v1
[Wat44] Watson, G. N., A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Related content

Powered by UNSILO

Densities of Short Uniform Random Walks

  • Jonathan M. Borwein (a1), Armin Straub (a2), James Wan (a1), Wadim Zudilin (a1) and Don Zagier (a3)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.