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

Abstract : In this paper we present a thorough study of transport, spectral and wave-function properties at the Anderson localization critical point in spatial dimensions $d = 3$, $4$, $5$, $6$. Our aim is to analyze the dimensional dependence and to asses the role of the $d\rightarrow \infty$ limit provided by Bethe lattices and tree-like structures. Our results strongly suggest that the upper critical dimension of Anderson localization is infinite. Furthermore, we find 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 properties and finite size scaling behavior approach by increasing $d$ the ones found for Bethe lattices: the critical state becomes an insulator characterized by Poisson statistics and corrections to the thermodynamics limit become logarithmic in $N$. In the conclusion, we present physical consequences of our results, propose connections with the non-ergodic delocalised phase suggested for the Anderson model on infinite dimensional lattices and discuss perspectives for future research studies.
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Journal articles
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Cited literature [35 references]

https://hal-cea.archives-ouvertes.fr/cea-01545377
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### Citation

Elena Tarquini, Giulio Biroli, Marco Tarzia. Critical properties of the Anderson localization transition and the high dimensional limit. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2016, 95, pp.094204 ⟨10.1016/S0370-1573(01)00098-9⟩. ⟨cea-01545377⟩

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