On Pólya's random walk constants

Robert E. Gaunt, Saralees Nadarajah, Tibor K. Pogány

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A celebrated result in probability theory is that a simple symmetric random walk on the d-dimensional lattice Zd is recurrent for d=1,2 and transient for  3. In this note, we derive a closed-form expression, in terms of the Lauricella function of type C, for the return probability for all  3. Previously, a closed-form formula had only been available for d=3.
Original languageEnglish
Title of host publicationProceedings of the American Mathematical Society
PublisherAmerican Mathematical Society
ISBN (Electronic)1088-6826
ISBN (Print)0002-9939
Publication statusAccepted/In press - 3 Jan 2024


  • Random walk
  • return probability
  • P´olya’s random walk constants
  • Lauricella function
  • Watson’s triple integrals
  • Laplace transform


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