Series studies of self-avoiding walks near the θ-points on 2D and 3D clusters at the percolation thresholds

Abstract

Enumerating all the N-stepped SAW configurations on the infinite percolation cluster of Monte Carlo generated bond diluted lattices (in dimension d=2 as well as in d=3) at the respective percolation thresholds, the thermally weighted average end-to-end distance (R_N) of self-interacting self-avoiding walks are determined. The configurationally averaged (R_N)^̅ (over different percolation clusters) are then fitted to a scaling form (R²_N)^̅ ~N^2vθ f(N^∅t), where t=(T-θ)/θ denotes the temperature interval away from the θ-point, v^θis the tricritical ( θ-point) size exponent, ∅is the crossover exponent and f is the scaling function. From the best fit, the values of θ, v^θ and ∅ are obtained for the 2D and 3D lattices considered. We find v^θ⋍ 0.74±0.02 and 0.60±0.02 for the tricritical exponents on the percolation clusters (at the percolation thresholds) in dimensions d=2 and d=3 respectively. We also find theta-temperature (θ) ⋍ 0.71±0.15, 1.25±0.3 and 0.5±0.15 for bond dilute square, triangular and simple cubic lattices respectively on the critical percolation clusters. Our scaling fit results for θ-point and the v^θ values for various percolating lattices are then compared with some theoretical (mean field-like) estimates.