TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.18.2.057
TI  - Universality of closed nested paths in two-dimensional percolation
PY  - 2025/02/18
UR  - https://scipost.org/SciPostPhys.18.2.057
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 18
IS  - 2
SP  - 057
A1  - Song, Yu-Feng
AU  - Jacobsen, Jesper Lykke
AU  - Nienhuis, Bernard
AU  - Sportiello, Andrea
AU  - Deng, Youjin
AB  - Recent work on percolation in $d=2$ [J. Phys. A: Math. Theor. 55, 204002 (2015)] 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 $k ≥ -1$, 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 $\ell ≥ 0$ 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.
ER  -

@Article{10.21468/SciPostPhys.18.2.057,
	title={{Universality of closed nested paths in two-dimensional percolation}},
	author={Yu-Feng Song and Jesper Lykke Jacobsen and Bernard Nienhuis and Andrea Sportiello and Youjin Deng},
	journal={SciPost Phys.},
	volume={18},
	pages={057},
	year={2025},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.18.2.057},
	url={https://scipost.org/10.21468/SciPostPhys.18.2.057},
}