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Journal of Physics Condensed Matter

Dynamic structure factor of percolation clusters at criticality

Abdelali Rahmani, C Benoit, Gérard Poussigue

Published Nov 20, 1995• 6 citations

Abstract

The dynamic structure factor of very large percolating clusters (bond and site), in two dimensions and three dimensions are calculated using the spectral moments method. Interactions are represented by the scalar model. Numerical results are presented and interpreted in terms of scaling arguments given by Alexander, Courtens and Vacher (ACV) in 1993. Concerning the q lambda <<1 limit, our results confirm and supplement previous numerical work and are in agreement with the scaling behaviour theoretically deduced by ACV. Concerning the q lambda >>1 limit, we have shown that the dynamic structure factor complies with the very nice asymptotic behaviour g(q, omega )=q y H(q lambda ( omega )) where the scaling function H(x) is of power-law form x - tau ' in agreement with the theory. However, our results indicate a scaling behaviour with exponents that differ from those deduced by theory. The values obtained for tau ' are 1.20 for the two-dimensional bond and site percolating lattices, and 1.00 for the three-dimensional percolating lattices. Comparisons with previous simulations are reported.

DOI: 10.1088/0953-8984/7/47/011

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The dynamic structure factor of very large percolating clusters (bond and site), in two dimensions and three dimensions are calculated using the spectral moments method. Interactions are represented by the scalar model. Numerical results are presented and interpreted in terms of scaling arguments given by Alexander, Courtens and Vacher (ACV) in 1993. Concerning the q lambda <<1 limit, our results confirm and supplement previous numerical work and are in agreement with the scaling behaviour theoretically deduced by ACV. Concerning the q lambda >>1 limit, we have shown that the dynamic structure factor complies with the very nice asymptotic behaviour g(q, omega )=q y H(q lambda ( omega )) where the scaling function H(x) is of power-law form x - tau ' in agreement with the theory. However, our results indicate a scaling behaviour with exponents that differ from those deduced by theory. The values obtained for tau ' are 1.20 for the two-dimensional bond and site percolating lattices, and 1.00 for the three-dimensional percolating lattices. Comparisons with previous simulations are reported.

The scaling behavior of the dynamical structure factor S(q,\ensuremath{\omega}) of percolating Heisenberg antiferromagnets is investigated in terms of a dynamic scaling argument and numerical calculations. It is found, based on the single-length-scaling postulate, that the asymptotic behavior of S(q,\ensuremath{\omega}) can be characterized by only two exponents: the dynamic exponent

{\mathit{z}}_{\mathit{a}}

and a new exponent y. This theoretical prediction is confirmed by direct numerical calculations. The values of the exponents

{\mathit{z}}_{\mathit{a}}

and y are determined for the case of d=2 percolating Heisenberg antiferromagnets.

Abstract Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.

The critical behaviour of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ξ, ξ being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed, and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by θ = λ = 1.

Random-walk simulations on square-lattice percolation clusters are performed for (a) all clusters at criticality; (b) the largest cluster at nominal criticality; and (c) the largest cluster at exact criticality for each realization (new method). The short-time correction-to-scaling exponent is obtained: \ensuremath{\omega}=-0.48 (for N

{<10}^{3}

Monte Carlo steps). The scaling exponent appears to be in better agreement with the recent Aharony-Stauffer conjecture for the spectral dimension (

{d}_{s}

=1.309) than with the Alexander-Orbach-Rammal-Toulouse conjecture (

{d}_{s}

=1.333).