Dimensionalities and other geometric critical exponents in percolation theory

A review is given of the studies of the dimensionality characteristics of percolation clusters. The purely geometric nature of a percolation phase transition and the great variety of the quantities exhibiting critical behavior make this geometric approach both informative and useful. In addition to the fractal dimensionality of a cluster and its subsets (such as the backbone, hull, and other dimensionalities), it is necessary to introduce additional characteristics. For example, the maximum velocity of propagation of excitations is determined by the chemical dimensionality of a cluster, and the critical behavior of the conductivity, diffusion coefficient, etc., is determined by spectral (or other related to it) dimensionalities. Scaling relationships between different dimensionalities, as well as relationships between dimensionalities and conventional critical exponents are discussed.

PACS: 64.60.Ak, 64.60.Fr (all)
DOI: 10.1070/PU1986v029n10ABEH003526
URL: https://ufn.ru/en/articles/1986/10/b/
Citation: Sokolov I M "Dimensionalities and other geometric critical exponents in percolation theory" Sov. Phys. Usp. 29 924–945 (1986)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Соколов И М «Размерности и другие геометрические критические показатели в теории протекания» УФН 150 221–255 (1986); DOI: 10.3367/UFNr.0150.198610b.0221

Cited by (141) Similar articles (20) ↓

B.I. Shklovskii, A.L. Éfros “Percolation theory and conductivity of strongly inhomogeneous media” 18 845–862 (1975)
B.M. Smirnov “Fractal clusters” 29 481–505 (1986)
L.M. Zelenyi, A.V. Milovanov “Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics” 47 749–788 (2004)
A.I. Olemskoi, I.A. Sklyar “Evolution of the defect structure of a solid during plastic deformation” 35 (6) 455–480 (1992)
V.V. Zosimov, L.M. Lyamshev “Fractals in wave processes” 38 347–384 (1995)
A.I. Olemskoi, A.Ya. Flat “Application of fractals in condensed-matter physics” 36 (12) 1087–1128 (1993)
Ya.B. Zel’dovich, S.A. Molchanov et al “Intermittency in random media” 30 353–369 (1987)
A.S. Mikhailov, I.V. Uporov “Critical phenomena in media with breeding, decay, and diffusion” 27 695–714 (1984)
A.A. Likal’ter “Gaseous metals” 35 (7) 591–605 (1992)
V.I. Alkhimov “The self-avoiding walk problem” 34 (9) 804–816 (1991)
V.I. Alkhimov “Excluded volume effect in statistics of self-avoiding walks” 37 527–561 (1994)
G.A. Smolenskii, R.V. Pisarev, I.G. Sinii “Birefringence of light in magnetically ordered crystals” 18 410–429 (1975)
G.M. Zaslavskii, B.V. Chirikov “Stochastic instability of non-linear oscillations” 14 549–568 (1972)
M.A. Anisimov “Investigations of critical phenomena in liquids” 17 722–744 (1975)
R. Folk, Yu. Holovatch, T. Yavorskii “Critical exponents of a three-dimensional weakly diluted quenched Ising model” 46 169–191 (2003)
E.Z. Meilikhov “Structural features, critical currents and current-voltage characteristics of high temperature superconducting ceramics” 36 (3) 129–151 (1993)
V.K. Vanag “Study of spatially extended dynamical systems using probabilistic cellular automata” 42 413–434 (1999)
V.G. Boiko, Kh.I. Mogel’ et al “Features of metastable states in liquid-vapor phase transitions” 34 (2) 141–159 (1991)
M.A. Anisimov, E.E. Gorodetskii, V.M. Zaprudskii “Phase transitions with coupled order parameters” 24 57–75 (1981)
Ya.B. Zel’dovich, A.L. Buchachenko, E.L. Frankevich “Magnetic-spin effects in chemistry and molecular physics” 31 385–408 (1988)

The list is formed automatically.