Nested closed paths in two-dimensional percolation

Abstract

Abstract 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 ∼ L − X NP , 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 ϕ 2 − 5 48 ϕ 2 / ( ϕ 2 − 2 3 ) with k = 2 cos(πϕ), 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.