Upper critical dimension of the Anderson transition
In plain words
For many phase transitions there is a number of spatial dimensions above which the exponents become simple and stop changing. For the localization transition of electrons in a random potential it is unknown whether such a dimension exists or is infinite.
Precise statement
For the orthogonal-class Anderson model on a d-dimensional hypercubic lattice, determine whether a finite upper critical dimension $d_u$ exists above which $\nu$ and the multifractal spectrum are d-independent (proposals include $d_u = 6$ and $d_u = \infty$), and how the $d \to \infty$ limit connects to the Anderson transition on the Bethe lattice and random regular graphs, whose critical scaling differs from ordinary finite-size scaling. An answer is $d_u$, or a demonstration that $d_u = \infty$ together with the large-d asymptotics of $\nu(d)$.
What would settle it
A controlled expansion around the infinite-dimensional (Bethe-lattice) limit, or numerics in $d = 4\text{ to }7$ with controlled corrections to scaling, showing whether exponents become d-independent at finite d.
Status in the literature
Unverified note
Numerics in $d = 3\ \text{to}\ 6$ were read as $d_{u} = \infty$ (Tarquini, Biroli and Tarzia, Physical Review B 95, 094204, 2017); random-matrix descriptions of the infinite-dimensional limit continue (Chen, Giraud, Gong and Lemarie, Physical Review B 110, 014210, 2024).