Journal of Physics A Mathematical and General

Electrical conductivity of finite-size percolation networks

Catalin D Mitescu, Mickaël Allain, E. Guyon, J. P. Clerc

Published Aug 1, 1982• 39 citations

Abstract

Calculating, by a Monte Carlo technique, the conductivities of small, two-dimensional, square, bond-percolation networks (4*4 to 50*50) at a limited number of probability values, from the critical region to full conductivity, the authors have been able to show, by an original interpolation technique, that in the critical region the conductivity function does indeed obey a universal, Fisher-type, finite-size scaling function. They further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length, even from calculations on relatively small networks.

DOI: 10.1088/0305-4470/15/8/030

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Journal of Physics A Mathematical and General

Electrical conductivity of finite-size percolation networks

Catalin D Mitescu, Mickaël Allain, E. Guyon, J. P. Clerc

Published Aug 1, 1982• 39 citations

Publisher

Abstract

Calculating, by a Monte Carlo technique, the conductivities of small, two-dimensional, square, bond-percolation networks (44 to 5050) at a limited number of probability values, from the critical region to full conductivity, the authors have been able to show, by an original interpolation technique, that in the critical region the conductivity function does indeed obey a universal, Fisher-type, finite-size scaling function. They further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length, even from calculations on relatively small networks.

DOI: 10.1088/0305-4470/15/8/030

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Critical exponent for 3-D percolation conductivity, revisited

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Journal de Physique LettresJan 1, 198349 citations

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{E}_{p},

the percolation probability

P,

and the probability

{W}_{n}

for the appearance of

n

percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions

L.

Using the HMCSM, we calculate the existence probability

{E}_{p}

and the percolation probability

P

for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of

{E}_{p}

and

P

in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that

{E}_{p}

and

P

have universal finite-size scaling functions. This implies that the critical

{E}_{p}

is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that

{W}_{n}

for site and bond percolation on sc lattices have universal finite-size scaling functions.