Anderson localization as a percolation transition

Anderson localization is a fundamental phenomenon in condensed matter physics that describes the suppression of wave propagation due to disorder-induced interference effects. The Bethe lattice solution is the only known exact solution to the Anderson localization problem. A recent proposal by Filoche and Mayboroda claims to predict the position of the mobility edge in Anderson localization for systems in finite dimensions. This framework, that they called ''Localization Landscape Theory", introduces a function called localization landscape whose inverse acts as an effective potential, identifying regions where quantum states are spatially confined. The main claim of Filoche and Mayboroda's work is that the Anderson transition occurs when the regions with effective potential energy lower than the eigenstate energy percolate through the lattice—thus interpreting Anderson localization as a percolation transition. In this work we derive and solve numerically the equations determining the transition according to the localization landscape percolation on the Bethe lattice, showing that this framework is not able to reproduce the critical properties of Anderson localization on a generic lattice.