Universality of closed nested paths in two-dimensional percolation

TL;DR

This paper derives an exact formula for the scaling of nested paths in 2D percolation, confirming their universal behavior through simulations and advancing understanding of critical phenomena.

Contribution

It provides an exact expression for the scaling exponent of nested paths in 2D percolation, replacing previous conjectures and confirming universality.

Findings

Exact formula for $X_{NP}(k)$ derived for $k \,\geq\, -1$

Probability distribution of nested paths scales with $L^{-1/4} (\,\ln L)^{\ell}$

Simulations confirm theoretical predictions and universality of nested path observables.

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 ${\rm E}(k^{\ell}) \sim L^{-X_{\rm NP}(k)}$, 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.