J. Phys. I France
Volume 3, Numéro 10, October 1993
Page(s) 2007 - 2016
DOI: 10.1051/jp1:1993228
J. Phys. I France 3 (1993) 2007-2016

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

K. Barat, S. N. Karmakar and B. K. Chakrabarti

Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700064, India

(Received 12 May 1993, accepted in final form 22 June 1993)

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 $\langle R_N \rangle$ 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 $\tau = (T - \theta)/\theta$ denotes the temperature interval away from the $\theta$-point, $\nu^\theta$ is the tricritical ( $\theta$-point) size exponent, $\phi$ is the crossover exponent and f is the scaling function. From the best fit, the values of $\theta$, $\nu^\theta$ and $\phi$ are obtained for the 2D and 3D lattices considered. We find $\nu^\theta \simeq 0.74\pm 0.02$ and $0.60\pm 0.02$ 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 $(\theta) \simeq0.71\pm0.15$, $1.25\pm0.3$ and $0.5\pm0.15$ for bond dilute square, triangular and simple cubic lattices respectively on the critical percolation clusters. Our scaling fit results for $\theta$-point and the $\nu^\theta$ values for various percolating lattices are then compared with some theoretical (mean field-like) estimates.

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