Series studies of self-avoiding walks near the ?-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 〈RN 〉 of self-interacting self-avoiding walks are determined. The configurationally averaged $\overline{\langle R_N \rangle}$ (over different percolation clusters) are then fitted to a scaling form $\overline{\langle R_N^2 \rangle}\sim N^{2\nu^\theta}\,f(N^\phi\tau)$, where τ= (T - θ)/θ denotes the temperature interval away from the θ-point, νθ is the tricritical (θ-point) size exponent, φ is the crossover exponent and f is the scaling function. From the best fit, the values of θ, νθ and φ are obtained for the 2D and 3D lattices considered. We find νθ≃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 νθ values for various percolating lattices are then compared with some theoretical (mean field-like) estimates.