  We analyze the top Lyapunov exponent of the product of sequences of two by
two matrices that appears in the analysis of several statistical mechanics
models with disorder: for example these matrices are the transfer matrices for
the nearest neighbor Ising chain with random external field, and the free
energy density of this Ising chain is the Lyapunov exponent we consider. We
obtain the sharp behavior of this exponent in the large interaction limit when
the external field is centered: this balanced case turns out to be critical in
many respects. 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
$\log$-H\"older continuous.
