Nested Closed Paths in Two-Dimensional Percolation

Abstract

For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path operator (NP) and thus a continuous family of one-point functions $W_k\equiv ⟨\mathcal{R}\cdot k^{\ell }⟩$, where $\ell$ is the number of independent nested closed paths surrounding the center, $k$ is a path fugacity, and $\mathcal{R}$ projects on configurations having a cluster connecting the center to the boundary. At criticality, we observe a power-law scaling $W_k\sim L^{X_{\mathrm{NP}}}$, with $L$ the linear system size, and we determine the exponent $X_{\mathrm{NP}}$ as a function of $k$. On the basis of our numerical results, we conjecture an analytical formula, $X_{\mathrm{NP}}(k)=\frac{3}{4}\ddot{I}{•}^2-\frac{5}{48}\ddot{I}{•}^2/(\ddot{I}{•}^2-\frac{2}{3})$ where $k=2\cos (\ddot{I}€\ddot{I}•)$, which reproduces the exact results for $k=0,1$ and agrees with the high-precision estimate of $X_{\mathrm{NP}}$ for other $k$ values. In addition, we observe that $W_2(L)=1$ for site percolation on the triangular lattice with any size $L$, and we prove this identity for all self-matching lattices.

5 pages, 5 figures plus supplemental material, 1 page 2 figures