Long time behavior of run-and-tumble particles in two dimensions

We study the long-time asymptotic behavior of the position distribution of a run-and-tumble particle (RTP) in two dimensions in the presence of translational diffusion and show that the distribution at a time t can be expressed as a perturbative series in (gamma t)(-1), where gamma(-1) is the persistence time of the RTP. We show that the higher order corrections to the leading order Gaussian distribution generically satisfy an inhomogeneous diffusion equation where the source term depends on the previous order solutions. The explicit solution of the inhomogeneous equation requires the position moments, and we develop a recursive formalism to compute the same. We find that the subleading corrections undergo shape transitions as the translational diffusion is increased.