  We introduce a new correlated percolation model on the $d$-dimensional
lattice $\mathbb{Z}^d$ called the random length worms model. Assume given a
probability distribution on the set of positive integers (the length
distribution) and $v \in (0,\infty)$ (the intensity parameter). From each site
of $\mathbb{Z}^d$ we start $\mathrm{POI}(v)$ independent simple random walks
with this length distribution. We investigate the connectivity properties of
the set $\mathcal{S}^v$ of sites visited by this cloud of random walks. It is
easy to show that if the second moment of the length distribution is finite
then $\mathcal{S}^v$ undergoes a percolation phase transition as $v$ varies.
Our main contribution is a sufficient condition on the length distribution
which guarantees that $\mathcal{S}^v$ percolates for all $v>0$ if $d \geq 5$.
E.g., if the probability mass function of the length distribution is $ m(\ell)=
c \cdot \ln(\ln(\ell))^{\varepsilon}/ (\ell^3 \ln(\ell)) 1[\ell \geq \ell_0] $
for some $\ell_0>e^e$ and $\varepsilon>0$ then $\mathcal{S}^v$ percolates for
all $v>0$. Note that the second moment of this length distribution is only
"barely" infinite. In order to put our result in the context of earlier results
about similar models (e.g., finitary random interlacements, loop percolation,
Poisson Boolean model, ellipses percolation, etc.), we define a natural family
of percolation models called the Poisson zoo and argue that the percolative
behaviour of the random length worms model is quite close to being "extremal"
in this family of models.
