Returns to the origin of the Pólya walk with stochastic resetting

Abstract

We consider the simple random walk (or Pólya walk) on the one-dimensional lattice subject to stochastic resetting to the origin with probability $r$ at each time step. The focus is on the joint statistics of the numbers ${\mathcal{N}}_t^{\times}$ of spontaneous returns of the walker to the origin and ${\mathcal{N}}_t^{\bullet}$ of resetting events up to some observation time $t$. These numbers are extensive in time in a strong sense: all their joint cumulants grow linearly in $t$, with explicitly computable amplitudes, and their fluctuations are described by a smooth bivariate large deviation function. A non-trivial crossover phenomenon takes place in the regime of weak resetting and late times. Remarkably, the time intervals between spontaneous returns to the origin of the reset random walk form a renewal process described in terms of a single `dressed' probability distribution. These time intervals are probabilistic copies of the first one, the `dressed' first-passage time. The present work follows a broader study, covered in a companion paper, on general nested renewal processes.

Figures (8)

• Figure 1: Example of a path of the Pólya walk on the one-dimensional lattice under stochastic resetting, generated by a simulation with $r=0.08$. The walk starts at the origin. It restarts afresh at the origin at each resetting event, figured by a dot. Spontaneous returns to the origin are figured by crosses.
• Figure 2: Sketch of the temporal events for the path of figure \ref{['fig:snap']}. Spontaneous returns to the origin of the walk are figured by crosses, resetting events by dots. The intervals of time between two crosses, ${\boldsymbol{\tau}}_1,{\boldsymbol{\tau}}_2,\dots$, have common distribution $\rho(\tau)$ (see (\ref{['eq:defrho']}), (\ref{['eq:fctngen']})). The intervals of time between two resettings, $\boldsymbol T_1,\dots,\boldsymbol T_4$, have the geometric distribution (\ref{['eq:geom']}). The last interval, $B_t$, represents the backward recurrence time, or age of the resetting process at time $t$, i.e., the time elapsed since the previous resetting event.
• Figure 3: Amplitudes $A^\bullet=r$, $A^\times$ and their sum $A$ entering the growth laws (\ref{['eq:meannb']}), (\ref{['eq:ncasy']}) and (\ref{['ncbasy']}) of $\langle{\mathcal{N}}_t^{\bullet}\rangle$, $\langle{\mathcal{N}}_t^{\times}\rangle$ and $\langle{\mathcal{N}}_t^{\times\bullet}\rangle$, plotted against the resetting probability $r$.
• Figure 4: The time intervals between spontaneous returns to the origin (crosses) of the reset random walk form a renewal process, described in terms of the 'dressed' probability distribution (\ref{['eq:dressed2']}). These time intervals are probabilistic copies of the first one, $\boldsymbol{T}_{0\to0}^{(\tt r)}$, the 'dressed' first-passage time. (Compare to figure \ref{['fig:nested']}.)
• Figure 5: Amplitude $A^\times$ (see (\ref{['eq:ncasy']}) and (\ref{['acave']})) and decay rate $\sigma$ (see (\ref{['eq:sigrat']})), plotted against the resetting probability $r$.