The Theta Point

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 R_N of SAW’s of N steps, i.e., N(N + 1) ~ R ^−d_N where d is the dimensionality of the space. The relationship also holds at the Theta point: N _ω(N + 1) ~ R ^−d_θ where R_θ is the radius of gyration of the interacting SAW’s at the θ-temperature and N_ω is the appropriately weighted polygon number. We show that a walk on the hull of the percolation clusters at the critical threshold P_c of the triangular lattice is identical to an interacting SAW and the critical properties of this walk are the θ-point critical properties.