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We prove a conditional theorem on the positivity of the Lyapunov exponent fora Schr\"odinger cocycle over a skew shift base with a cosine potential and thegolden ratio as frequency. For coupling below 1, which is the threshold forHerman's subharmonicity trick, we formulate three conditions on the Lyapunovexponent in a finite but large volume and on the associated large deviationestimates at that scale. Our main results demonstrate that these finite-sizeconditions imply the positivity of the infinite volume Lyapunov exponent. Thispaper shows that it is possible to make the techniques developed for the studyof Schr\"odinger operators with deterministic potentials, based on largedeviation estimates and the avalanche principle, effective.

Effective multi-scale approach to the Schr\"odinger cocycle over a skew shift base