The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin

Mordechai Gruda, Ofer Biham*, Eytan Katzav, Reimer Kühn

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We present analytical results for the joint probability distribution P ( T FR = t , S = s ) of first return (FR) times t and of the number of distinct sites s visited by a random walk (RW) on a one dimensional lattice before returning to the origin. The RW on a one dimensional lattice is recurrent, namely the probability to return to the origin is P R = 1 . However the mean ⟨ T FR ⟩ of the distribution P ( T FR = t ) of FR times diverges. Similarly, the mean ⟨ S ⟩ of the distribution P ( S = s ) of the number of distinct sites visited before returning to the origin also diverges. The joint distribution P ( T FR = t , S = s ) provides a formulation that controls these divergences and accounts for the interplay between the kinetic and geometric properties of FR trajectories. We calculate the conditional distributions P ( T FR = t | S = s ) and P ( S = s | T FR = t ) . Using moment generating functions and combinatorial methods, we find that the conditional expectation value of FR times of trajectories that visit s distinct sites is E [ T FR | S = s ] = 2 3 ( s 2 + s + 1 ) , and the variance is Var ( T FR | S = s ) = 4 45 ( s − 1 ) ( s + 2 ) ( s 2 + s − 1 ) . We also find that in the asymptotic limit, the conditional expectation value of the number of distinct sites visited by an RW that first returns to the origin at time t = 2 n is E [ S | T FR = 2 n ] ≃ π n , and the variance is Var ( S | T FR = 2 n ) ≃ π ( π 3 − 1 ) n . These results go beyond the important recent results of Klinger et al (2022 Phys. Rev. E 105 034116), who derived a closed form expression for the generating function of the joint distribution, but did not go further to extract an explicit expression for the joint distribution itself. The joint distribution provides useful insight on the efficiency of random search processes, in which the aim is to cover as many sites as possible in a given number of steps. A further challenge will be to extend this analysis to higher-dimensional lattices, where the FR trajectories exhibit complex geometries.

Original languageEnglish
Article number013203
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2025
Issue number1
DOIs
StatePublished - 1 Jan 2025

Bibliographical note

Publisher Copyright:
© 2025 The Author(s). Published on behalf of SISSA Medialab srl by IOP Publishing Ltd.

Keywords

  • Catalan number
  • Dyck paths
  • first return time
  • random walk
  • recurrence

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