Critical exponents from scaling with neglect of cutoffs

The neglect of cutoff dependences enables the derivation of scaling relations by pure dimensional analysis. Scaling relations for systems in confined volumes are utilized to provide the Imry et al. interdimensional relations for static and dynamic critical exponents. Illustrations are given for $\nu$, $\beta$, and $z$. For $n=0\mathrm{}$, the Flory value for $\nu$ is obtained, while for $n=1\mathrm{}$ the errors in the three-dimensional values of $\nu$ and $\beta$ are 5 and 10%, respectively, from our simple algebraic recursion relation. Our approach is based upon asymptotic dimensional arguments for systems governed by Landau-Ginzburg-type free-energy functionals, and this formulation also enables the separation of relevant and irrelevant variables near the critical point. The utility of the scaling theory is illustrated by application to the problem of the description of the electronic structure of disordered materials where traditional renormalization-group methods yields runaway solutions. The present methods (neglecting cutoffs) yield nontrivial information concerning conductivity and density-of-states exponents near the mobility edge.