“Thermodynamical formalism” for an infinite hierarchy of fractal resistor networks

A thermodynamical formalism for resistor networks is developed in order to extract full information on the multifractal scaling structure. We introduce a matrix representation and study the moments of the voltage distribution Z̃(β) = ∑i=1N|Vi|β α N -F̄(β) where N is the number of resistors. We find a generic phase transition at β = βc = -1. Also, we develop a transfer matrix technique which determines all even positive moments. The thermodynamical formalism is applied to the Hilfer-Blumen hierarchy of generalized Sierpinski gasket fractal networks, and the crossover from fractal to lattice behavior is studied. At this crossover we find a sharp phase transition in the second moment (β = 2).