We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as a function of the correlation strength. The correlations are generated using a discrete version of the Fourier filtering method. We consider two different metrics to set the length scales over which the correlations decay, showing that the percolation thresholds are highly sensitive to such system details. By contrast, we verify that the fractal dimension d_{f} is a universal quantity and unaffected by the choice of metric. We also show that for weak correlations, its value coincides with that for the uncorrelated system. In two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength, approaching d_{f}→2. The onset of this change does not seem to be determined by the extended Harris criterion.

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We present a new method for exact enumeration of self-avoiding walks on critical percolation clusters. It can handle very long walks by exploiting the clusters' low connectivity and self-similarity. We have implemented the method in 2D and used it to enumerate walks of more than 1000 steps with over 10 170 conformations. The exponents ν and γ, governing the scaling behavior of the end-to-end distance and the number of configurations, as well as the connectivity constant µ could thus be determined with unprecedented accuracy. The method will help answering long-standing questions regarding this particular problem and can be used to check and gauge other methods, analytical and numerical. It can be adapted to higher dimensions and might also be extended to similar systems.

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We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters' fractal nature. We enumerate walks of over 10 4 steps, far more than has ever been possible. The scaling exponent ν for the end-to-end distance turns out to be smaller than previously thought and appears to be the same on the backbones as on full clusters. We find strong evidence against the widely assumed scaling law for the number of conformations and propose an alternative, which perfectly fits our data.PACS numbers: 64.60.al, 64.60.De The self-avoiding walk (SAW) [1] is a fundamental model in statistical mechanics and crucial for our understanding of the scaling behavior of polymers [2]. Asymptotically, it is characterized by universal exponents, which are related to the critical exponents of spin systems and assumed to describe long, flexible polymers in good solvent condition. While much is known about SAWs on regular lattices, their behavior in disordered environments, such as porous rocks or biological cells, is less understood. The paradigmatic model for such systems are SAWs on critical percolation clusters [3,4]. Here the walks can only visit a random fraction of sites, whose concentration is equal to the percolation threshold of the lattice. This critical concentration may not be realistic, but it represents an important limiting case, and the effect of the critical clusters' fractal structure is particularly intriguing [5].One usually considers quenched disorder averages, here denoted by square brackets: On each disorder realization ("cluster"), one takes the average over all walk conformations of length N . Each such conformational average contributes equally to the disorder average. It is assumed that the average number of conformations, [Z], and their mean squared end-to-end distance, R 2 , follow asymptotic scaling laws similar to those for normal SAWs:γ and ν are universal scaling exponents, d f is the SAW's fractal (Hausdorff) dimension, and µ is a lattice dependent effective connectivity constant. While the effect of the fractal disorder on γ and µ is still very controversial, there is convincing evidence that ν is different than on regular lattices [6]. However, there is uncertainty concerning the actual value despite a considerable amount of work dedicated to the system. Analytical works have yielded conflicting results [7][8][9], while accuracy and reliability of numerical investigations have been poor due to modest system and sample sizes. In most numerical studies (see for instance [10][11][12][13][14] We recently developed a new algorithm for exact enumeration of SAWs on two-dimensional critical percolation clusters [21], which we have now generalized to higher dimensions. By making use of the clusters' fractal properties, it overcomes the exponential increase in computation time that usually affects exact enumeration methods. Walks of over 10 4 steps are now accessible, permitting ...