On the surface properties of two-dimensional percolation clusters

Queiroz, S L A de

doi:10.1088/0305-4470/28/13/002

Cited by 4 publications

(9 citation statements)

References 20 publications

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“…Thus we have a correction-to-scaling exponent equal to x(θ k ) for the two-point percolation probability. Another correction is due to the leading irrelevant operator with scaling dimension −1 at the surface [17]. This explains the value, close to 1, of the effective correction exponent ω for two-point percolation, since the amplitude of the first correction is small.…”

Section: Discussionmentioning

confidence: 87%

“…The conformal aspects of the critical percolation problem in finite geometries have been extensively studied in [15], following the work of [16]. Conformal invariance has been also verified in a transfer-matrix calculation of the surface percolation exponent, using the gap-exponent relation [17]. The critical behaviour at surface and corners has been considered in [18] where the conduction problem is addressed briefly.…”

Section: Introductionmentioning

confidence: 99%

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Percolation and conduction in restricted geometries

Lajkó

Turban²

2000

J. Phys. A: Math. Gen.

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The finite-size scaling behaviour for percolation and conduction is studied in two-dimensional triangular-shaped random resistor networks at the percolation threshold. The numerical simulations are performed using an efficient star-triangle algorithm. The percolation exponents, linked to the critical behaviour at corners, are in good agreement with the conformal results. The conductivity exponent, t ′ = ζ/ν, is found to be independent of the shape of the system. Its value is very close to recent estimates for the surface and bulk conductivity exponents.

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Section: Discussionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Percolation and conduction in restricted geometries

Lajkó

Turban²

2000

J. Phys. A: Math. Gen.

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“…Building up the transfer matrix involves the analysis of connectivity properties of adjacent columns of occupied and empty sites. For the present case of animals on strips with FBC, this is a straightforward extension of earlier work on percolation and animals with periodic boundary conditions (PBC) [8,9] and percolation with FBC [14]. The resulting matrix is rather sparse, owing to restrictions imposed by connectivity [8]: for L = 10 on the square lattice, for instance, only 4.2% of the possible combinations of adjacent column states are allowed.…”

Section: Model and Calculational Proceduresmentioning

confidence: 72%

“…This is to be compared e.g. to similar extrapolations for percolation with FBC [14] where usually one can pinpoint a much narrower band of values of ω within which fluctuations are minimised.…”

Section: Squarementioning

confidence: 99%

Surface crossover exponent for branched polymers in two dimensions

Queiroz

1995

J. Phys. A: Math. Gen.

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Transfer-matrix methods on finite-width strips with free boundary conditions are applied to lattice site animals, which provide a model for randomly branched polymers in a good solvent. By assigning a distinct fugacity to sites along the strip edges, critical properties at the special (adsorption) and ordinary transitions are assessed. The crossover exponent at the adsorption point is estimated as φ = 0.505 ± 0.015, consistent with recent predictions that φ = 1/2 exactly for all space dimensionalities.

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“…. has been investigated elsewhere [12]. For our present purposes the relevant comparison is between the columns of Table 1, which shows the soundness of the proposed scheme.…”

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confidence: 99%

Connectivity-dependent properties of diluted sytems in a transfer-matrix description

Queiroz

Stinchcombe

1998

Phys. Rev. E

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We introduce a new approach to connectivity-dependent properties of diluted systems, which is based on the transfer-matrix formulation of the percolation problem. It simultaneously incorporates the connective properties reflected in non-zero matrix elements and allows one to use standard random-matrix multiplication techniques. Thus it is possible to investigate physical processes on the percolation structure with the high efficiency and precision characteristic of transfer-matrix methods, while avoiding disconnections. The method is illustrated for two-dimensional site percolation by calculating (i) the critical correlation length along the strip, and the finite-size longitudinal DC conductivity: (ii) at the percolation threshold, and (iii) very near the pure-system limit.The transfer-matrix (TM) approach to percolation was pioneered by Derrida and coworkers [1]. By analysing the possible combinations of adjacent column states made up of occupied and unoccupied sites (or bonds), and the allowed connections among the latter and to the arbitrary origin of a two-dimensional (d = 2) strip, it was possible to write the TM on the basis of such column states. The key element in this formulation was the fact that, from the very structure of the TM, repeated multiplication is tantamount to the simultaneous generation of all possible connected configurations that span the strip, each with its proper probabilistic weight. Thus the probability of connection to the origin, whose exponential decay is governed by the correlation length, is asymptotically given exactly by the largest eigenvalue of the TM. The correlation length could then be used in a phenomenological renormalisation calculation [2], which gave very accurate results for critical parameters such as the percolation threshold and correlation-length exponent ν. It was not clear, however, how one could take advantage of such a direct and elegant scheme to investigate properties other than the decay of the probability of connection to the origin. The obvious alternative, of building up successive columns by occupying (or not) each individual site independently, runs into the problem of disconnections, which is severely aggravated on a strip geometry. Up to now, the usual solution has been to generate configurations site by site, and study quantities which do not depend on keeping connectivity along the strip, e.g. the moments of the distribution of clusters [3] for percolation in d = 2 and 3. A clever way to get round the effects of disconnections for random resistor-insulator networks at percolation was introduced [4] by generating individual elements on long strips (or bars, in d = 3) with free edges. By imposing a fixed voltage drop across the strip, it was possible to invoke TM concepts in a step-by-step evaluation of the transverse conductivity, for which longitudinal disconnections of the resistor structure are irrelevant. Finally, in superconductor-resistor networks at the percolation threshold of superconducting elements, disconnections are in fact responsib...

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On the surface properties of two-dimensional percolation clusters

Cited by 4 publications

References 20 publications

Percolation and conduction in restricted geometries

Percolation and conduction in restricted geometries

Surface crossover exponent for branched polymers in two dimensions

Connectivity-dependent properties of diluted sytems in a transfer-matrix description

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