A ug 2 00 1 Universality of the critical conductance distribution in various dimensions

We study numerically the metal-insulator transition in the Anderson model on various lattices with dimension 2 < d ≤ 4 (bifractals and Euclidian lattices). The critical exponent ν and the critical conductance distribution are calculated. We confirm that ν depends only on the spectral dimension. The other parameters-critical disorder , critical conductance distribution and conductance cummulants-depend also on lattice topology. Thus only qualitative comparison with theoretical formulae for dimension dependence of the cummulants is possible. It is commonly accepted, though not proved, that metal-insulator transitions (MIT) can be described by one-parameter scaling theory [1]. The critical exponent ν which describes the divergence of correlation length at MIT depends only on the system dimension for a chosen universality class. Microscopic details of models do not affect it. This was confirmed by numerical analysis of quasi-one-dimensional (Q1D) systems [2]. Theoretical dependence of ν on dimension d = 2 + ε was found in [3, 4] for small ε. Numerically, ν(ε) was studied on bifractals [5]. The conductance g was originally chosen as the order parameter in the scaling theory [1]. Soon it became clear, that the absence of self-averaging of g in the critical region must be taken into account [6, 7]. The shape of the critical conductance distribution P (g) in 3D models was numerically analysed in detail [8, 9, 10, 11, 12]. Contrary to the critical exponent, P (g) is not universal. Its shape depends not merely on the dimension [10] and physical symmetry [11] but also on boundary conditions [12] and even anisotropy [13]. Nevertheless, for a given physical model the mean conductance and resistance follow one parameter scaling [14]. Analytical theory of MIT is restricted to systems with dimension close to the lower critical dimension: 2 + ε with ε ≪ 1 [15]. In spite of predicted non-universality of higher order conductance cummulants δg n ,