Recent work on percolation in $d=2$ [J. Phys. A {\bf 55} 204002] introduced an operator that gives a weight $k^{\ell}$ to configurations with $\ell$ `nested paths' (NP), i.e. disjoint cycles surrounding the origin, if there exists a cluster that percolates to the boundary of a disc of radius $L$, and weight zero otherwise. It was found that ${\rm E}(k^{\ell}) \sim L^{-X_{\rm NP}(k)}$, and a formula for $X_{\rm NP}(k)$ was conjectured. Here we derive an exact result for $X_{\rm NP}(k)$, valid for $k \ge -1$, replacing the previous conjecture. We find that the probability distribution ${\rm P}_\ell (L)$ scales as $ L^{-1/4} (\ln L)^\ell [(1/\ell!) \Lambda^\ell]$ when $\ell \geq 0$ and $L \gg 1$, with $\Lambda = 1/\sqrt{3} \pi$. Extensive simulations for various critical percolation models confirm our theoretical predictions and support the universality of the NP observables.
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List of changes
A few typo's corrected (lines 482, 507, 557) clarifying word 'exactly' added (to avoid interpretation 'at least') in line 395 added reference to subsequent work that goes beyond ours. (arXiv-2401.15904)
