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

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 $\langle T_{FR}\rangle$ of the distribution $P(T_{FR}=t)$ of first return times diverges. Similarly, the mean $\langle S\rangle$ 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 first return trajectories. We calculate the conditional distributions $P(T_{FR}=t|S=s)$ and $P(S=s|T_{FR}=t)$. We find that the conditional expectation value of first return times of trajectories that visit s distinct sites is ${\mathbb E}[T_{FR}|S=s]=\frac{2}{3}(s^2+s+1)$, and the variance is $Var(T_{FR}|S=s)=\frac{4}{45}(s-1)(s+2)(s^2+s-1)$. 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 $t=2n$ is ${\mathbb E}[S|T_{FR}=2n] \simeq \sqrt{\pi n}$, and the variance is $Var(S|T_{FR}=2n) \simeq \pi\left(\frac{\pi}{3}-1\right)n$. These results go beyond the important recent results of Klinger et al. [{\it Phys. Rev. E} {\bf 105}, 034116 (2022)], 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.