LETTER TO THE EDITOR Infinite set of exponents describing physics on fractal networks

Abstract. The generalised resistance between connected points a distance 4 apart on fractal networks of non-linear ( V- I") resistors scales as Lc'"'. We show that l ( n) for, " =-a,-1, 0-, O+, 1 and 03, describes physically relevant geometrical properties and dC/du S 0. For percolating clusters we give approximants for 5 for-03 < a < 03 in 2-6 dimensions. For a < 0 a family of solutions to Kirchhoffs equations exists, reminiscent of metastable states in spin glasses. Random resistor fractal networks have been the subject of much fruitful research recently. Different properties of such systems are found to probe different critical exponents, or fractal dimensionalities. Choosing two terminals a Euclidean distance L apart, and putting a voltage- V between them, generates a current I = V / R, and the resistance scales as R ( L) - L'R (the resistance of each bond is r = 1). Current flows only via backbone bonds, whose number MB scales with the fractal dimensionality of the backbone D,: MB- L D ~. The characterisation of the network also invplves the Lsc, singly connected bonds, which carry the full current I, with Lsc- Lrsc (these determine the low temperature spin correlation functions (Coniglio 198 1,1982, Aharony et a1 1984) and give bounds for the elastic response of networks (Kantor-and Webman 1984)), the length of the minimal path between the terminals Lmin- L g m i n (related to the rate of propagation of a flow front through the cluster (Stanley 1986)) and other physically relevant subgroups described below. The study of percolating resistor networks recently led to the identification of several infinite sets of exponents, relevant to their physical properties. Rammal et a1 (1985) considered resistance fluctuations arising from microscopic noise in each resistor ri in the network. The kth cumulant of R ( L) then scales as ( R k) , - Lx(k). If the current in the ith bond is Ii = Iii, then (Rammal et a1 1985