## Abstract

A celebrated result in probability theory is that a simple symmetric random walk on the

*d*-dimensional lattice**Z**is recurrent for^{d}**and transient for***d*=1,2*d*≥**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*d*≥**3**. Previously, a closed-form formula had only been available for**.***d*=3Original language | English |
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Title of host publication | Proceedings of the American Mathematical Society |

Publisher | American Mathematical Society |

ISBN (Electronic) | 1088-6826 |

ISBN (Print) | 0002-9939 |

Publication status | Accepted/In press - 3 Jan 2024 |

## Keywords

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