Abstract
We exploit the relationship between the limit of the n → 0 of the n-vector model and self-avoiding walks (SAW) to relate the number of closed polygons, N of N + 1 links to the radius of gyration RN of SAW’s of N steps, i.e., N(N + 1) ~ R N −d where d is the dimensionality of the space. The relationship also holds at the Theta point: N ω(N + 1) ~ R θ −d where Rθ is the radius of gyration of the interacting SAW’s at the θ-temperature and Nω is the appropriately weighted polygon number. We show that a walk on the hull of the percolation clusters at the critical threshold Pc of the triangular lattice is identical to an interacting SAW and the critical properties of this walk are the θ-point critical properties.
Published Version
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