Localization for Random CMV Matrices

The paper proves Anderson localization and dynamical localization in expectation for i.i.d. CMV matrices on any compact arc disjoint from an exceptional set D of at most three points, by combining positivity/continuity of Lyapunov exponents, uniform LDT, a uniform Craig–Simon bound, and a key regularity theorem (Theorems 1–2, with the exceptional set described in §4.3) . The candidate solution reaches the same conclusions via a Furstenberg-based route: it constructs a finite exceptional set (≤3 points) by analyzing shared boundary fixed points of SU(1,1) transfer matrices, proves positivity of the Lyapunov exponent off that set, and then invokes standard 1D i.i.d. localization machinery to deduce AL and EDL. While the model’s exceptional-set discussion contains a minor inaccuracy (it suggests two-point invariance forces z=1, whereas the paper allows other possibilities like {1,−1}), and it omits the paper’s uniform LDT/Craig–Simon ingredients needed for a fully detailed CMV proof, its overall logic and end results align with the paper’s theorems. Thus both are correct, but by different methods.