Nested closed paths in two-dimensional percolation

For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path (NP) operator and thus a continuous family of one-point functions W-k < R . k(l)>, where l is the number of independent (i.e., non-overlapping) nested closed paths surrounding the center, k is a path fugacity, and R projects on configurations having a cluster connecting the center to the boundary. At criticality, we observe a power-law scaling W-k similar to L-XNP with L the linear system size, and we determine the exponent X-NP as a function of k. On the basis of our numerical results, we conjecture an analytical formula, X-NP(k) = 3/4 phi(2) - 5/48 phi(2)/(phi(2) - 2/3) with k = 2 cos(pi phi), which reproduces the exact results for k = 0, 1 and agrees with the high-precision estimate of X-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.

Automated summary

In this study of percolation on a domain with the topology of a disc, a nested-path operator was introduced, connecting a continuous family of one-point functions to the structure's center. The power-law scaling behavior at criticality was observed, and an analytical formula for the exponent as a function of path fugacity was proposed based on numerical results. In addition, an identity for site percolation on the triangular lattice was discovered and proven to hold for self-matching lattices of any size.