### Speaker

### Description

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 $t_l$ of the last visit to the

origin. Interestingly, the cumulative distribution of all three quantities are the

same and given by Arcsine function. In this paper, we study distribution of

these three times $t_m$, $t_r$ and $t_l$ 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 a delta function part and a non-delta function

part. Interestingly, we find that the distributions of $t_m$ and tr 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 $t_l$, only

the non-delta function part is the same as 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.