Condensation transition in the late-time position of a run-and-tumble particle

We study the position distribution $P(\vec{R},N)$ of a run-and-tumble particle (RTP) in arbitrary dimension $d$, after $N$ runs. assume that constant speed $v>0$ during each running phase is independently drawn from probability $W(v)$ and direction chosen isotropically tumbling. The clearly isotropic, $P(\vec{R},N)\to P(R,N)$ where $R=|\vec{R}|$. show that, under certain conditions on $d$ for large $N$, condensation transition occurs at some critical value $R=R_c\sim O(N)$ located deviation regime $P(R,N)$. For $RR_c$ typically dominated by `condensate', i.e., single run subsumes finite fraction total displacement (supercritical condensed phase). Focusing family distributions $W(v)=\alpha(1-v/v_0)^{\alpha-1}/v_0$, parametrized $\alpha>0$, we $P(R,N)\sim \exp\left[-N\psi_{d,\alpha}(R/N)\right]$ compute exactly rate function $\psi_{d,\alpha}(z)$ any $\alpha$. manifests itself as singularity this $R=R_c$ its order depends continuously also condensate $R>R_c$. Finally, model when duration $T$ RTP, instead number runs, fixed. Our analytical predictions confirmed numerical simulations, performed using constrained Markov chain Monte Carlo technique, precision $\sim 10^{-100}$.