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We consider random products of $SL(2, \mathbb{R})$ matrices that depend on aparameter in a non-uniformly hyperbolic regime. We show that if the dependenceon the parameter is monotone then almost surely the random product has upper(limsup) Lyapunov exponent that is equal to the value prescribed by theFurstenberg Theorem (and hence positive) for all parameters, but the lower(liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set ofparameters of zero Hausdorff dimension. As a byproduct of our methods, weprovide a purely geometrical proof of Spectral Anderson Localization fordiscrete Schr\"odinger operators with random potentials (including theAnderson-Bernoulli model) on a one dimensional lattice.

Parametric Furstenberg Theorem on Random Products of $SL(2, \mathbb{R})$ matrices