Noncrossing run-and-tumble particles on a line

We study active particles performing independent run-and-tumble motion on an infinite line with velocities v_{0}σ(t), where σ(t)=±1 is a dichotomous telegraphic noise with constant flipping rate γ. We first consider a single particle in the presence of an absorbing wall at x=0 and calculate the probability that it has survived up to time t and is at position x at time t. We then consider two particles with independent telegraphic noises and compute exactly the probability that they do not cross up to time t. In contrast with the case of passive (Brownian) particles this problem of two run-and-tumble particles (RTPs) cannot be reduced to a single RTP with an absorbing wall. Nevertheless, we are able to compute exactly the probability of no crossing of two independent RTPs up to time t and find that it decays at large time as t^{-1/2} with an amplitude that depends on the initial condition. We show that this amplitude vanishes if one extrapolates the starting distance between the two particles to a negative value-similarly to the Milne extrapolation length in neutron scattering.