Regular Expansion for the Characteristic Exponent of a Product of 2 × 2 Random Matrices

We consider a product of 2 × 2 random matrices which appears in the physics literature in the analysis of some 1D disordered models. These matrices depend on a parameter 𝜖 > 0 and on a positive random variable Z. Derrida and Hilhorst (J. Phys. 16(12), 2641, 1983, § 3) conjecture that the corresponding characteristic exponent has a regular expansion with respect to 𝜖 up to — and not further — an order determined by the distribution of Z. We give a rigorous proof of that statement. We also study the singular term which breaks that expansion.