Uniform estimates for random matrix products and applications

For certain natural families of topologies, we study continuity and stability of statistical properties of random walks on linear groups over local fields. We extend large deviation results known in the Archimedean case to non-Archimedean local fields and also demonstrate certain large deviation estimates for heavy tailed distributions unknown even in the Archimedean case. A key technical result, which may be of independent interest, establishes lower semi-continuity for the gap between the first and second Lyapunov exponents. As applications, we are able to obtain a key technical step towards a localization proof for heavy tailed Anderson models (the full proof appearing in a companion article), and show continuity/stability (taking the geometric data as input) of various statistical data associated to hyperbolic surfaces.