  We study the position distribution of a single active Brownian particle (ABP)
on the plane. We show that this distribution has a compact support, the
boundary of which is an expanding circle. We focus on a short-time regime and
employ the optimal fluctuation method (OFM) to study large deviations of the
particle position coordinates $x$ and $y$. We determine the optimal paths of
the ABP, conditioned on reaching specified values of $x$ and $y$, and the large
deviation functions of the marginal distributions of $x$, and of $y$. These
marginal distributions match continuously with "near tails" of the $x$ and $y$
distributions of typical fluctuations, studied earlier. We also calculate the
large deviation function of the joint $x$ and $y$ distribution $P(x,y,t)$ in a
vicinity of a special "zero-noise" point, and show that $\ln P(x,y,t)$ has a
nontrivial self-similar structure as a function of $x$, $y$ and $t$. The joint
distribution vanishes extremely fast at the expanding circle, exhibiting an
essential singularity there. This singularity is inherited by the marginal $x$-
and $y$-distributions. We argue that this fingerprint of the short-time
dynamics remains there at all times.
