Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization in two and three dimensions

For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions d=2,3, we study numerically the statistical properties of the renormalized on-site energies epsilon and of the renormalized hoppings V as a function of the linear size L. We find that the renormalized on-site energies epsilon remain finite in the localized phase in d=2,3 and at criticality (d=3), with a finite density at epsilon=0 and a power-law decay 1/epsilon(2) at large vertical bar epsilon vertical bar. For the renormalized hoppings in the localized phase, we find: ln V-L similar or equal to-L xi(loc)+L(omega)u, where xi(loc) is the localization length and u a random variable of order one. The exponent omega is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension 1+(d-1), with omega(d=2)=1/3 and omega(d=3)similar or equal to 0.24. At criticality (d=3), the statistics of renormalized hoppings V is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure rho(typ)similar or equal to 1 for the exponent governing the typical decay ln V-L similar or equal to-rho(typ) ln L, in agreement with previous numerical measures of alpha(typ)=d+rho(typ)similar or equal to 4 for the singularity spectrum f(alpha) of individual eigenfunctions. We also present numerical results concerning critical surface properties.