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The phenomenon of upper critical dimensionality d_c2 has been studied fromthe viewpoint of the scaling concepts. The Thouless number g(L) is not the onlyessential variable in scale transformations, because there is the secondparameter connected with the off-diagonal disorder. The investigation of theresulting two-parameter scaling has revealed two scenarios, and the switchingfrom one to another scenario determines the upper critical dimensionality. Thefirst scenario corresponds to the conventional one-parameter scaling and ischaracterized by the parameter g(L) invariant under scale transformations whenthe system is at the critical point. In the second scenario, the Thoulessnumber g(L) grows at the critical point as L^{d-d_c2}. This leads to violationof the Wegner relation s=\nu(d-2) between the critical exponents forconductivity (s) and for localization radius (\nu), which takes the forms=\nu(d_c2-2). The resulting formulas for g(L) are in agreement with thesymmetry theory suggested previously [JETP 81, 925 (1995)]. A more rigorousversion of Mott's argument concerning localization due topological disorder hasbeen proposed.Comment: PDF, 7 pages, 6 figure

Scaling near the upper critical dimensionality in the localization theory