I will consider the top Lyapunov exponent of the product of IID two by two random matrices that appear in the study of various disordered statistical mechanics models. The focus will be on the singular behavior of this Lyapunov exponent and I will mostly restrict to a special case that is naturally considered «critical» both from a mathematical and physical perspective. The presentation will be mostly on the mathematical aspects with a part explaining what the Furstenberg theory (on products of random matrices) says on the Lyapunov exponent I consider, and a part devoted to sharp estimates recently obtained on this exponent. Ultimately, the key questions boil down to understanding the invariant measures of certain specific Markov chains. The recent results have been obtained in collaborations with Orphée Collin, Rafael Greenblatt and Yueyun Hu.
