  We investigate analytically the distribution tails of the area A and
perimeter L of a convex hull for different types of planar random walks. For N
noninteracting Brownian motions of duration T we find that the large-L and A
tails behave as $\mathcal{P}\left(L\right)\sim e^{-b_{N}L^{2}/DT}$ and
$\mathcal{P}\left(A\right)\sim e^{-c_{N}A/DT}$, while the small-$L$ and $A$
tails behave as $\mathcal{P}\left(L\right)\sim e^{-d_{N}DT/L^{2}}$ and
$\mathcal{P}\left(A\right)\sim e^{-e_{N}DT/A}$, where $D$ is the diffusion
coefficient. We calculated all of the coefficients ($b_N, c_N, d_N, e_N$)
exactly. Strikingly, we find that $b_N$ and $c_N$ are independent of N, for
$N\geq 3$ and $N \geq 4$, respectively. We find that the large-L (A) tails are
dominated by a single, most probable realization that attains the desired L
(A). The left tails are dominated by the survival probability of the particles
inside a circle of appropriate size. For active particles and at long times, we
find that large-L and A tails are given by $\mathcal{P}\left(L\right)\sim
e^{-T\Psi_{N}^{\text{per}}\left(L/T\right)}$ and $\mathcal{P}\left(A\right)\sim
e^{-T\Psi_{N}^{\text{area}}\left(\sqrt{A}/T\right)}$ respectively. We calculate
the large deviation functions $\Psi_N$ exactly and find that they exhibit
multiple singularities. We interpret these as dynamical phase transitions of
first order. We extended several of these results to dimensions $d>2$. Our
analytic predictions display excellent agreement with existing results that
were obtained from extensive numerical simulations.
