Universality of closed nested paths in two-dimensional percolation

TL;DR

This work derives an exact expression for the two-dimensional NP exponent $X_{\rm NP}(k)$ (valid for $k\ge -1$) and shows that the NP observables exhibit universal power-law scaling across multiple critical percolation models. It establishes precise relations linking NP to the nested-loop and NL exponents, via duality on self-matching lattices, and connects polychromatic NP to the watermelon exponent. The authors also determine the universal scaling form for the NP-number distribution $\mathbb{P}_{\ell}(L)$, including a leading logarithmic factor $\Lambda=\pi/\sqrt{3}$ and a logarithmic growth of the conditional NP count $N\sim (3/(8\pi))\ln L$, supported by extensive Monte Carlo simulations. Special cases at $k=0,1,2$ are discussed, with $X(0)=1/4$, $X(1)=5/48$, and $W_2(L)=1$ on triangulation lattices, while logarithmic corrections appear near $k=-1$, suggestive of a BKT-like transition. Overall, the results reinforce the universality of NP observables in 2D critical percolation and lay groundwork for extensions to Potts models and related systems.

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!) Λ^\ell]$ when $\ell \geq 0$ and $L \gg 1$, with $Λ= 1/\sqrt{3} π$. Extensive simulations for various critical percolation models confirm our theoretical predictions and support the universality of the NP observables.

Figures (15)

• Figure 1: Two matching pairs constructed from the square lattice. For the left two figures (M$_1$), half of the elementary faces are chosen, and diagonals are added either to faces in the chosen set (left) or to those in its complement (right). The generated pair of matching lattices are isomorphic and are denoted Sq$_6$. For the right two figures (M$_2$), none of the squared faces is chosen, and the matching pair corresponds to the original square lattice (Sq) and a square lattice with both nearest- and next-nearest neighboring interactions (Sq$_8$).
• Figure 2: Illustration of the bond-to-site transformation. Each edge in the bond-percolation problem is transformed into a vertex (blue dot) in the site-percolation problem, and two sites are taken to be neighboring if the corresponding bonds are adjacent. This example maps bond percolation on the self-dual lattice (BSq) onto site percolation on the self-matching lattice (Sq$_6$).
• Figure 3: Configurations contributing to the one-point function of the polychromatic 4-arm operator. A bijective transformation described in the text, a$\leftrightarrow$b and b$\leftrightarrow$c, adds or removes a radial cluster boundary, here indicated in bold white.
• Figure 4: Examples of a set of site configurations on the triangular lattice with $\ell=2$ nested closed paths. The central site is neutral (white), the occupied (empty) sites are represented as red (green) sites, and the sites on the fixed boundary are marked gray. The map $P_1$, associated with the first NP, leads to (a) $\leftrightarrow$ (b) and (c) $\leftrightarrow$ (d), while $P_2$ leads to (a) $\leftrightarrow$ (c) and (b) $\leftrightarrow$ (d). For a given statistical weight $k$, all the four configurations contribute to the one-point NL function $W_\textsc{nl}(k)$, with a total amount $1+k+k+k^2 = (k+1)^2$. By comparison, only (a) contributes to the one-point NP function $W_\textsc{np}(k)$ with an amount $k^2$.
• Figure 5: Union Jack lattice with the center site (denoted by the red star) on different sublattices (UJ$_4$ and UJ$_8$). The coordination number of the center site is 4 for the left and 8 for the right.