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In this paper we present a thorough study of transport, spectral andwave-function properties at the Anderson localization critical point in spatialdimensions $d = 3$, $4$, $5$, $6$. Our aim is to analyze the dimensionaldependence and to asses the role of the $d\rightarrow \infty$ limit provided byBethe lattices and tree-like structures. Our results strongly suggest that theupper critical dimension of Anderson localization is infinite. Furthermore, wefind that the $d_U=\infty$ is a much better starting point compared to $d_L=2$to describe even three dimensional systems. We find that critical propertiesand finite size scaling behavior approach by increasing $d$ the ones found forBethe lattices: the critical state becomes an insulator characterized byPoisson statistics and corrections to the thermodynamics limit becomelogarithmic in $N$. In the conclusion, we present physical consequences of ourresults, propose connections with the non-ergodic delocalised phase suggestedfor the Anderson model on infinite dimensional lattices and discussperspectives for future research studies.

Critical properties of the Anderson localization transition and the high dimensional limit