** IL SEMINARIO SI TERRA' ALLE ORE 16 ** I will discuss a certain infinite product of random, positive 2×2 matrices appearing in the exact solution of some 1 and 1+1 dimensional disordered models in statistical mechanics, which depends on a deterministic real parameter ε and a random real parameter with distribution μ. For a large class of μ, we prove a prediction by B. Derrida and H. J. Hillhorst (1983) that the leading Lyapunov exponent behaves like C ε^2α in the limit ε→0, where α ∈ (0,1) is determined by μ. The proof is made possible by a contractivity argument which makes it possible to control the error involved in using an approximate stationary distribution similar to the original proposal, along with some refinements in the estimates obtained using that distribution. A limiting procedure gives a continuum process whose leading Lyapunov estimate admits an exact formula, which also allows us to reformulate part of the argument by McCoy and Wu for the presence of an essential singularity in the free energy of the two-dimensional Ising model with columnar disorder in a form which is closely related to the results obtained for the random matrix product, but which does not yet provide a proof.