Obstructions to the regularity of the Lyapunov exponents for non-compact random Schr\"odinger cocycles

In this paper, we present a class of random Schr"odinger cocycles showing that, for random cocycles with non-compact support, the presence of certain finite moment conditions is essential for establishing a specific modulus of continuity of the Lyapunov exponent. In particular, H"older continuity of the Lyapunov exponent requires an exponential moment condition.