Abstract: The talk will focus on the top Lyapunov exponent of the transfer matrix for the nearest neighbor Ising chain with random external field. This Lyapunov exponent coincides with the free energy of the Ising chain with random external field, but it also plays a central role in the analysis of the two dimensional Ising model with columnar disorder and of the quantum chain with transverse random field.

The aim of the talk is to present a recent result (obtained with R. L. Greenblatt) on the sharp behavior of the Lyapunov exponent in the large interaction limit when the external field is centered: this «balanced» case turns out to be critical in many respects. We will explain this aspect by giving a presentation that includes the non critical case. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only log-Hölder continuous.