  Recent work on two-dimensional percolation [arXiv:2102.07135] introduced an
operator that counts the number of nested paths (NP), which is the maximal
number of disjoint concentric cycles sustained by a cluster that percolates
from the center to the boundary of a disc of diameter $L$. Giving a weight $k$
to each nested path, with $k$ a real number, the one-point function of the NP
operator was found to scale as $ L^{-X_{\rm NP}(k)}$, with a continuously
varying exponent $X_{\rm NP}(k)$, for which an analytical formula was
conjectured on the basis of numerical result. Here we revisit the NP problem.
We note that while the original NPs are monochromatic, i.e. all on the same
cluster, one can also consider polychromatic nested paths, which can be on
different clusters, and lead to an operator with a different exponent. The
original nested paths are therefore labeled with MNP. We first derive an exact
result for $X_{\rm MNP}(k)$, valid for $k \ge -1$, which replaces the previous
conjecture. Then we study the probability distribution $\mathbb{P}_{\ell}$ that
$\ell \geq 0$ NPs exist on the percolating cluster. We show that
$\mathbb{P}_{\ell}(L)$ scales as $ L^{-1/4} (\ln L)^\ell [(1/\ell!)
\Lambda^\ell]$ when $L \gg 1$, with $\Lambda = 1/\sqrt{3} \pi$, and that the
mean number of NPs, conditioned on the existence of a percolating cluster,
diverges logarithmically as $\kappa \ln L$, with $\kappa =3/8\pi$. These
theoretical predictions are confirmed by extensive simulations for a number of
critical percolation models, hence supporting the universality of the NP
observables.
