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Universality of closed nested paths in two-dimensional percolation

by Yu-Feng Song, Jesper Lykke Jacobsen, Bernard Nienhuis, Andrea Sportiello, Youjin Deng

Submission summary

Authors (as registered SciPost users):

Jesper Lykke Jacobsen · Bernard Nienhuis

Abstract

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 $\mathbb{E}(k^{\ell}) \sim L^{-\X{np}(k)}$, and a formula for $\X{np}(k)$ was conjectured. Here we derive an exact result for $\X{np}(k)$, valid for $k \ge -1$, replacing the previous conjecture. We find that the probability distribution $\mathbb{P}_\ell (L)$ 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.