Spectral dimension and conductivity exponent of the percolating cluster

Issue		J. Physique Lett. Volume 44, Number 23, décembre 1983


Page(s)		925 - 928
DOI		10.1051/jphyslet:019830044023092500

J. Physique Lett. 44, 925-928 (1983)
DOI: 10.1051/jphyslet:019830044023092500

Spectral dimension and conductivity exponent of the percolating cluster

M. Daoud

Laboratoire Léon Brillouin, CEN Saclay, 91191 Gif-sur-Yvette Cedex, France

Abstract
We discuss a recent conjecture by Alexander and Orbach about the spectral dimension d of a percolating cluster. We consider the transformation from percolation to backbone by removing the dangling ends, and those exponents which remain invariant in this transformation. We suggest d = (4( d + 2) - 4 η)/(3(d + 2) - 2 η) where d is the dimension of space, leading to the former conjecture if one supposes η = 0. This allows for an evaluation of the conductivity exponent t.

Résumé
Nous discutons une conjecture récente d'Alexander et Orbach sur la dimension spectrale d d'un amas percolant. Nous envisageons la transformation permettant de passer de la percolation au « squelette » par suppression des bras morts. Certains exposants sont invariants par cette transformation. Nous suggérons d = (4(d + 2) - 4 η)/(3(d + 2) - 2 η), en accord avec la conjecture citée si on suppose η = 0. Ceci nous permet de calculer l'exposant de conductivité t. Les résultats sont en accord raisonnable avec les simulations numériques.

PACS
0550 - Lattice theory and statistics: Ising problems.

Key words
electrical conductivity -- percolation -- conductivity exponent -- percolating cluster -- spectral dimension -- backbone -- dangling ends

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