Scale-free enumeration of self-avoiding walks on critical percolation clusters

Fricke, Niklas; Janke, Wolfhard

doi:10.1209/0295-5075/99/56005

Cited by 7 publications

(13 citation statements)

References 41 publications

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“…Chaingrowth Monte Carlo methods may allow for more than a hundred steps [15][16][17][18][19], but they add statistical uncertainty and the danger of biased results [20]. 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.…”

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Asymptotic Scaling Behavior of Self-Avoiding Walks on Critical Percolation Clusters

Fricke

Janke

2014

Phys. Rev. Lett.

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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 ...

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Asymptotic Scaling Behavior of Self-Avoiding Walks on Critical Percolation Clusters

Fricke

Janke

2014

Phys. Rev. Lett.

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“…It would be very interesting to also study the behavior of self-avoiding walks on long-range correlated percolating clusters. For the uncorrelated case, there exist exact enumeration techniques that enable treatment of extremely long walks [55,56]. While this approach becomes more demanding for an increasing correlation strength (a → 0), it should still be applicable to some extent.…”

Section: Conclusion and Prospectsmentioning

confidence: 99%

Scaling laws for random walks in long-range correlated disordered media

Fricke¹,

Zierenberg

Marenz³

et al. 2017

Condens. Matter Phys.

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We study the scaling laws of diffusion in two-dimensional media with long-range correlated disorder through exact enumeration of random walks. The disordered medium is modelled by percolation clusters with correlations decaying with the distance as a power law, r −a , generated with the improved Fourier filtering method. To characterize this type of disorder, we determine the percolation threshold p c by investigating cluster-wrapping probabilities. At p c , we estimate the (sub-diffusive) walk dimension d w for different correlation exponents a. Above p c , our results suggest a normal random walk behavior for weak correlations, whereas anomalous diffusion cannot be ruled out in the strongly correlated case, i.e., for small a.

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“…Fortunately though, we recently discovered that the fractal structure of percolation clusters offers a way to circumvent this problem [58,59]. The actual implementation of our method is rather complicated and will not be explained here in detail, but the basic ideas are fairly simple.…”

Section: Exact Enumerationmentioning

confidence: 99%

“…The resulting estimates of the exponent ν p c [65] are compiled in the second last line of table 1, where previously obtained results [71][72][73][74][75][76] are listed also for comparison. With our new exact enumeration scheme, we have so far only analyzed the two-dimensional case [58,59]. Our result for ν p c in the last line of table 1 is based on walks with up to N = 1 000 steps averaged over 200 000 percolation clusters at p c (working here with the full cluster, not the backbone).…”

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confidence: 99%

Polymers in disordered environments

Blavatska¹,

Fricke²,

Janke³

2014

Condens. Matter Phys.

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A brief review of our recent studies aiming at a better understanding of the scaling behaviour of polymers in disordered environments is given. The main emphasis is on a simple generic model where the polymers are represented by (interacting) self-avoiding walks and the disordered environment by critical percolation clusters. The scaling behaviour of the number of conformations and their average spatial extent as a function of the number of monomers and the associated critical exponents γ and ν are examined with two complementary approaches: numerical chain-growth computer simulations using the PERM algorithm and complete enumerations of all possible polymer conformations employing a recently developed very efficient exact counting method.

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Scale-free enumeration of self-avoiding walks on critical percolation clusters

Cited by 7 publications

References 41 publications

Asymptotic Scaling Behavior of Self-Avoiding Walks on Critical Percolation Clusters

Asymptotic Scaling Behavior of Self-Avoiding Walks on Critical Percolation Clusters

Scaling laws for random walks in long-range correlated disordered media

Polymers in disordered environments

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