Random 2D Linear Cocycles II: Statistical Properties

The paper proves, for random 2D cocycles with both singular and invertible letters, a stretched-exponential large deviations bound of the form exp(−c n^{1/3}) and a central limit theorem for log||A_n|| for Lebesgue-a.e. cocycle, via an explicit stationary measure on P^1, uniform ergodicity of an extended Markov operator Q̄, and parameter elimination; see Theorems 1.1 and 1.2 and their parametric counterparts Theorems 3.2 and 3.4 . A key technical point is that the natural additive observable φ(i, v̂)=log(||A_i v||/||v||) is unbounded below in the presence of singular letters, so one cannot directly invoke classical spectral-gap/analytic-perturbation methods (Le Page/Bougerol) that assume invertibility and regularity; the paper instead leverages uniform ergodicity of Q̄ on L∞ and an abstract LDT for bounded observables together with parameter elimination to obtain the n^{1/3} rate . The candidate solution asserts exponential LDT and a CLT by applying Le Page/Bougerol under SIP and a Doeblin regeneration argument, but this overlooks that the t-twisted transfer operator loses uniform minorization (weights can be arbitrarily small when the projective state approaches kernels of singular letters) and that φ is not uniformly bounded—precisely the obstacles the paper circumvents with its sub-exponential scheme. The model also claims generic positivity of the variance without the detailed parameter arguments given in the paper's CLT (Theorem 3.4) . Hence the paper’s argument is sound and appropriately scoped; the model overreaches by applying tools that do not directly accommodate singular matrices.