# Generalised `Arcsine' laws for run-and-tumble particle in one dimension

## TL;DR

This paper extends the classical Arcsine laws to a one-dimensional run-and-tumble particle, deriving exact distributions for key times related to maximum displacement and visits, revealing new similarities and differences with Brownian motion.

## Contribution

It provides the first exact distributions for the times of maximum, last visit, and positive duration for a run-and-tumble particle, generalizing Arcsine laws to a non-Markovian process.

## Key findings

- Distributions of $t_m$ and $t_r$ are identical with equal initial velocity probabilities.
- Distributions of $t_ ext{last}$ differ from $t_m$ and $t_r$, with only the non-delta part matching.
- Explicit joint distributions of maximum displacement and its occurrence time are derived.

## Abstract

The 'Arcsine' laws of Brownian particles in one dimension describe distributions of three quantities: the time $t_m$ to reach maximum position, the time $t_r$ spent on the positive side and the time $t_\ell$ of the last visit to the origin. Interestingly, the cumulative distribution of all the three quantities are same and given by Arcsine function. In this paper, we study distribution of these three times $t_m,~t_r$ and $t_\ell$ in the context of single run-and-tumble particle in one dimension, which is a simple non-Markovian process. We compute exact distributions of these three quantities for arbitrary time and find that all three distributions have delta function part and a non-delta function part. Interestingly, we find that the distributions of $t_m$ and $t_r$ are identical (reminiscent of the Brownian particle case) when the initial velocities of the particle are chosen with equal probability. On the other hand, for $t_\ell$, only the non-delta function part is same with the other two. In addition, we find explicit expressions of the joint distributions of the maximum displacement and the time at which this maxima occurs. We verify all our analytical results through numerical simulations.