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