We calculate the a.c. frequency response of Sierpinski-gasket networks, in which the bonds consist of resistors R (or of impedances Z h ) and all nodes are connected to the circuit ground by identical capacitors C (or by impedances Z v ). The resulting complex, size-dependent admittance between any of the «principal» nodes and the circuit ground can be accurately described at all frequencies less than 1/RC by a finite-size scaling function whose exponents are combinations of the fractal dimension d f and the spectral or «fracton» dimension d s of the Sierpinski gasket. The response function also bears a striking similarity to experimental observations of the a.c. response of a random mixture of conducting and insulating particles Nous avons calcule la reponse en frequence en courant alternatif pour un reseau du type Sierpinski dans lequel les liens sont soit des resistances R (ou des impedances Z h ) et ou tous les nœuds sont relies a la terre par l'intermediaire de capacites C identiques (ou d'impedances Z v ). Pour toutes les frequences plus petites que 1/RC, l'admittance complexe resultante entre chacun des nœuds «principaux» et la terre peut etre exprimee avec precision au moyen d'une fonction d'echelle avec effet de taille finie, tous les exposants de cette fonction etant des combinaisons des dimensions fractale d f et spectrale d s du tamis de Sierpinski. La dependance en frequence de la fonction de reponse presente une tres forte ressemblance avec celle d'un melange aleatoire de particules conductrices et isolantes
[1]
B. Derrida,et al.
Renormalisation groups with periodic and aperiodic orbits
,
1983
.
[2]
B. Mandelbrot.
FRACTAL ASPECTS OF THE ITERATION OF z →Λz(1‐ z) FOR COMPLEX Λ AND z
,
1980
.
[3]
B. Mandelbrot,et al.
Solvable Fractal Family, and Its Possible Relation to the Backbone at Percolation
,
1981
.
[4]
Eytan Domany,et al.
Solutions to the Schrödinger equation on some fractal lattices
,
1983
.
[5]
M. J. Stephen.
Mean-field theory and critical exponents for a random resistor network
,
1978
.
[6]
S. Alexander,et al.
Density of states on fractals : « fractons »
,
1982
.
[7]
B. Southern,et al.
Scaling and density of states of fractal lattices from a generating function point of view
,
1983
.
[8]
R. Rammal.
Nature of eigenstates on fractal structures
,
1983
.
[9]
J. Straley,et al.
The ant in the labyrinth: diffusion in random networks near the percolation threshold
,
1980
.
[10]
A. Aharony.
Anomalous Diffusion on Percolating Clusters
,
1983
.
[11]
A. Tremblay,et al.
Chaotic scaling trajectories and hierarchical lattice models of disordered binary harmonic chains
,
1983
.