The Alexander–Orbach Conjecture is Corrected

Abstract

Using the Einstein relation [Formula: see text] the new critical exponents for regular diffusion are introduced [Formula: see text] where t is the conductivity critical exponents, g = -0.6 is the Hall coefficient critical exponents above the threshold for two-component composite. This result allows us to correct the Alexander–Orbach conjecture1 that an effective dimensionality of the random walk, is d w = 2+(t-g)/ν by changing β on g and obtaining a universal fraction dimension d s = 4/3 in two and three-dimensional spaces. The Alexander and Orbach suggestion is nothing else but a rigorous relation for dynamic exponents of conductivity, the Hall and diffusion coefficients, similar to those for the phase-transition static critical exponents. It allows us to verify our value of fractal dimension of the backbone [Formula: see text] instead of customary [Formula: see text] for three-dimensional space and also to calculate the Hall coefficient critical exponent and the backbone fractal dimension in high dimension spaces.