Densities of Short Uniform Random Walks

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

Canadian journal of mathematics, Tome 64 (2012) no. 5, pp. 961-990

Résumé

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.

Détail

DOI : 10.4153/CJM-2011-079-2

Mots-clés : 33C20, 60G50, 34M25, 44A10, 05A19, 11F11, random walks, hypergeometric functions, Mahler measure

Bibliographie

[Aka09] [Aka09] Akatsuka, H., Zeta Mahler measures. J. Number Theory 129(2009), no. 11, 2713–2734. Google Scholar | DOI

[BB10] [BB10] Bailey, D. H. and Borwein, J. M., Hand-to-hand combat with thousand-digit integrals. J. Computational Science, 2010, Google Scholar | DOI

[BBBG08] [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, 5203–5231. Google Scholar

[BB91] [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, 691–701. Google Scholar | DOI

[BB98] [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. Google Scholar

[BBG94] [BBG94] Borwein, J. M., Borwein, P. B., and Garvan, F., Some cubic modular identities of Ramanujan. Trans. Amer. Math. Soc. 343(1994), no. 1, 35–47. Google Scholar | DOI

[BNSW11] [BNSW11] Borwein, J. M., Nuyens, D., Straub, A., and Wan, J., Some arithmetic properties of random walk integrals. Ramanujan J. 26(2011), no. 1, 109–132. Google Scholar | DOI

[Boy81] [Boy81] Boyd, D. W., Speculations concerning the range of Mahler's measure. Canad. Math. Bull. 24(1981), no. 4, 453–469. Google Scholar | DOI

[BS11] [BS11] Borwein, J. M. and Straub, A., Log-sine evaluations of Mahler measures. J. Aust Math. Soc., to appear. arxiv:1103.3893. Google Scholar

[BSW11] [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. Google Scholar

[BZ92] [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, 519–526. Google Scholar | DOI

[BZB08] [BZB08] Borwein, J. M., Zucker, I. J., and Boersma, J., The evaluation of character Euler double sums. Ramanujan J. 15(2008), no. 3, 377–405. Google Scholar | DOI

[CZ10] [CZ10] Chan, H. H. and Zudilin, W., New representations for Apéry-like sequences. Mathematika 56(2010), 107–117. Google Scholar | DOI

[Cra09] [Cra09] Crandall, R. E., Analytic representations for circle-jump moments. PSIpress, 21 nov 09: E, http://www.perfscipress.com/papers/Analytic Wn psipress.pdf. Google Scholar

[DM04] [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, 351–354. Google Scholar | DOI

[DM07] [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, 157–194. Google Scholar | DOI

[Fet63] [Fet63] Fettis, H. E., On a conjecture of Karl Pearson. In: Rider anniversary volume, Defense Technical Information Center, Belvoir, VA, 1963, pp. 39–54. Google Scholar

[Fin05] [Fin05] Finch, S., Modular forms on SL2(Z). http://algo.inria.fr/csolve/frs.pdf Google Scholar

[FS09] [FS09] Flajolet, P. and Sedgewick, R., Analytic combinatorics. Cambridge University Press, Cambridge, 2009. Google Scholar

[Hör90] [Hör90] Hörmander, L., The analysis of linear partial differential operators. I. Second ed. Springer, Berlin, 1990. Google Scholar

[Hug95] [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. Google Scholar

[Inc26] [Inc26] Ince, E. L., Ordinary differential equations. Green and Co., London, 1926. Google Scholar

[Klu05] [Klu05] Kluyver, J. C., A local probability problem. In: Royal Netherlands Academy of Arts and Sciences, Proceedings, 8 I, 1905, pp. 341–350. Google Scholar

[Luk69] [Luk69] Luke, Y. L., The special functions and their approximations. Vol. 1, Academic Press, New York-London, 1969. Google Scholar

[ML86] [ML86] Misra, O. P. and Lavoine, J. L., Transform analysis of generalized functions. North-Holland Mathematical Studies, 119, North-Holland Publishing Co., Amsterdam, 1986. Google Scholar

[Nes03] [Nes03] Nesterenko, Y. V.. Integral identities and constructions of approximations to zeta-values. J. Théor. Nombres Bordeaux 15(2003), no. 2, 535–550. Google Scholar | DOI

[Pea05a] [Pea05a] Pearson, K., The problem of the random walk. Nature 72(1905), no. 1867, 342. Google Scholar

[Pea05b] [Pea05b] Pearson, K., The problem of the random walk. Nature 72(1905), no. 1865, 294. Google Scholar

[Pea06] [Pea06] Pearson, K., A mathematical theory of random migration. In: Drapers company research memoirs, Biometric Series, III, Cambridge University Press, Cambridge, 1906. Google Scholar

[PWZ96] [PWZ96] Petkovsek, M., Wilf, H., and Zeilberger, D., A = B. Peters, A. K., Wellesley, MA, 1996. Google Scholar

[Ray05] [Ray05] Rayleigh, The problem of the random walk. Nature 72(1905), no. 1866, 318. Google Scholar

[Rog09] [Rog09] Rogers, M. D., New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/. Ramanujan J. 18(2009), no. 3, 327–340. Google Scholar | DOI

[RVTV04] [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, 473–494. Google Scholar | DOI

[SC67] [SC67] Selberg, A. and Chowla, S., On Epstein's zeta-function. J. Reine Angew. Math. 227(1967), 86–110. Google Scholar | DOI

[Tit39] [Tit39] Titchmarsh, E. C., The theory of functions. Second ed., Oxford University Press, 1939. Google Scholar

[Ver04] [Ver04] Verrill, H. A., Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations. arxiv:math/0407327v1 Google Scholar

[Wat44] [Wat44] Watson, G. N., A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944. Google Scholar