Sharp localization on the first supercritical stratum for Liouville frequencies

We establish Anderson localization for Schr\"odinger operators with even analytic potentials on the first supercritical stratum for Liouville frequencies in the sharp regime : L(ω,E)>β(ω)>0, κ(ω,E)=1\, with κ(ω,E) being Avila's acceleration. This paper builds on the large deviation measure estimate and complexity bound scheme, originally developed for Diophantine frequencies by Bourgain, Goldstein and Schlag {BG,BGS1,BGS2}, and the improved complexity bounds in {HS1}. Additionally, it strengthens the large deviation estimates for weak Liouville frequencies in {HZ}. We also introduce new ideas to handle Liouville frequencies in a sharp way.