Analyticity of the Lyapunov exponents of random products of quasi-periodic cocycles

The paper proves that, for random products of quasi-periodic cocycles, the top Lyapunov exponent λ_+(p) depends real-analytically on the probability vector p when the top exponent is simple, and extends this to all exponents under simplicity of the full spectrum; this is stated as Theorem 1 and Corollary 2 and proved via a uniform-convergence method built on a quantitative fiber-contraction estimate (Proposition 5) and a holomorphic polynomial-operator scheme )_I, without invoking spectral-gap perturbation theory . By contrast, the model’s proof invokes a Ruelle/transfer-operator spectral gap on the product space T^ℓ × P(R^3) and applies Kato’s perturbation theory. This spectral gap is not established and is generally obstructed by the base being an isometry (translations), so the claimed Doeblin–Fortet inequality and Kato-analyticity for L_{z,t} are unsubstantiated in this setting. The paper’s argument avoids these pitfalls and is coherent and complete; hence paper correct, model wrong.