Universality of closed nested paths in two-dimensional percolation

Recent work on percolation in $d=2$ [J. Phys. A {\bf 55} 204002] introduced an operator that gives a weight $k^{\mathrm{\ell }}$ to configurations with $\mathrm{\ell }$ `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 $\mathrm{E}(k^{\mathrm{\ell }})\sim L^{-X_{\mathrm{N}\mathrm{P}}(k)}$, and a formula for $X_{\mathrm{N}\mathrm{P}}(k)$ was conjectured. Here we derive an exact result for $X_{\mathrm{N}\mathrm{P}}(k)$, valid for $k\ge -1$, replacing the previous conjecture. We find that the probability distribution ${\mathrm{P}}_{\mathrm{\ell }}(L)$ scales as $L^{-1\mathrm{/}4}(\ln L{)}^{\mathrm{\ell }}[(1\mathrm{/}\mathrm{\ell }!){\mathrm{\Lambda }}^{\mathrm{\ell }}]$ when $\mathrm{\ell }\ge 0$ and $L\gg 1$, with $\mathrm{\Lambda }=1\mathrm{/}\sqrt{3}\pi$. Extensive simulations for various critical percolation models confirm our theoretical predictions and support the universality of the NP observables.