Random Walks in Multidimensional Spaces, Especially on Periodic Lattices

Abstract

Previous article Random Walks in Multidimensional Spaces, Especially on Periodic LatticesElliot W. MontrollElliot W. Montrollhttps://doi.org/10.1137/0104014PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Georg Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann., 84 (1921), 149–160 MR1512028 CrossrefGoogle Scholar[2] W. H. McCrea and , F. J. W. Whipple, Random paths in two and three dimensions, Proc. Roy. Soc. Edinburgh, 60 (1940), 281–298 MR0002733 0027.33903 CrossrefGoogle Scholar[3] William Feller, An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons Inc., New York, N.Y., 1950xii+419 MR0038583 0039.13201 Google Scholar[4] C. Domb, On multiple returns in the random-walk problem, Proc. Cambridge Philos. Soc., 50 (1954), 586–591 MR0063596 0056.12602 CrossrefISIGoogle Scholar[5] A. Dvoretzky and , P. Erdös, Some problems on random walk in space, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950., University of California Press, Berkeley and Los Angeles, 1951, 353–367 MR0047272 0044.14001 Google Scholar[6] F. G. Foster and , J., I Good, On a generalization of Pólya's random-walk theorem, Quart. J. Math., Oxford Ser. (2), 4 (1953), 120–126 MR0055617 0050.13903 CrossrefGoogle Scholar[7] K. L. Chung and , W. H. J. Fuchs, On the distribution of values of sums of random variables, Mem. Amer. Math. Soc., 1951 (1951), 12– MR0040610 0042.37502 Google Scholar[8] G. N. Watson, Three triple integrals, Quart. J. Math., Oxford Ser., 10 (1939), 266–276 MR0001257 0022.33202 CrossrefGoogle Scholar[9] Elliott W. Montroll, Theory of the vibration of simple cubic lattices with nearest neighbor interactions, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley and Los Angeles, 1956, 209–246 MR0084266 0075.22302 Google Scholar[10A] Hermann Weyl, Mean Motion, Amer. J. Math., 60 (1938), 889–896 MR1507355 CrossrefGoogle Scholar[10B] Hermann Weyl, Mean Motion. II, Amer. J. Math., 61 (1939), 143–148 MR1507367 CrossrefGoogle Scholar[11] K. Pearson, The problem of the random walk, Nature, 72 (1905), 602– CrossrefGoogle Scholar[12] Lord Rayleigh, Scientific papers, , 6 (1920), Google Scholar[13] J. C. Kluyver, A local probability problem, Proceedings of the Section of Science of Koninklijke Academie van Wetenschappen te Amsterdam, 8 (1905), 341–350 Google Scholar[14] J. Arthur Greenwood and , David Durand, The distribution of length and components of the sum of n random unit vectors, Ann. Math. Statist., 26 (1955), 233–246 MR0069414 CrossrefISIGoogle Scholar[15] Aurel Wintner, Upon a Statistical Method in the Theory of Diophantine Approximations, Amer. J. Math., 55 (1933), 309–331 MR1506967 CrossrefGoogle Scholar[16] H. B. Rosenstock and , G. F. Newell, Vibrations of a simple cubic lattice I, J. Chem. Phys., 21 (1953), 1607–1608 10.1063/1.1699307 CrossrefISIGoogle Scholar Previous article FiguresRelatedReferencesCited byDetails Random Walk on Lattices with Uncertain TrapsHerbert B. Rosenstock12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 27, No. 3AbstractPDF (642 KB) Volume 4, Issue 4| 1956Journal of the Society for Industrial and Applied Mathematics History Submitted:17 July 1956Published online:10 July 2006 InformationCopyright © 1956 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0104014Article page range:pp. 241-260ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics