Classical localization problem: a survey
Zoey Zhou
TL;DR
The paper investigates classical localization phenomena arising from quantum network models in symmetry class C by mapping quantum transport to history‑dependent random walks and percolation–type geometries. It develops Cardy’s reduction to a classical trail formalism, analyzes two‑dimensional Manhattan and Lorentz‑mirror constructions, and applies enhancement and combinatorial enumeration techniques to obtain rigorous localization results. Notably, near criticality all trajectories are almost surely finite in the Manhattan setting, and on even‑width cylinders the maximal horizontal displacement grows only polynomially with the cylinder width, establishing a robust classical analogue of quantum localization phenomena. Collectively, the work advances understanding of universal localization mechanisms across planar and cylindrical geometries and provides a concrete, rigorous testing ground for percolation‑based methods in low dimensions.
Abstract
We survey classical localization problems arising from quantum network models in symmetry class C and their mappings to history-dependent random walks on directed lattices. We describe how localization versus delocalization of trajectories can be analysed using percolation methods and combinatorial enumeration of path intersection patterns. In particular, we review results establishing almost sure finiteness of trajectories for parameters near criticality and polynomial bounds on the confinement length in cylindrical geometries.