| Abstract: |
| We consider one-parameter families of random circle diffeomorphisms $(g_{E,\\bar{\\omega}})$ for which the unperturbed map $g_{0,\\bar{0}}$ has a parabolic fixed point, and the dependence on the parameter $E$ is monotone. Under reasonable assumptions, we show that the rotation number $\\rho(E)$ exhibits Lifshitz tails: doubly-exponential decay with exponent $-(2k-1)/2k$, $ \\lim_{E \\downarrow 0} \\frac{\\ln(-\\ln(\\rho(E) - \\rho(0)))}{\\ln(E)} = -\\frac{2k-1}{2k}. $ The exponent is determined by the passage time through a parabolic bottleneck. A full rotation requires on the order of $E^{-(2k-1)/2k}$ consecutive small perturbations, and the probability of such a streak decays exponentially as a function of its length. As a corollary, we provide a purely dynamical proof of the classical Lifshitz tails result for the one-dimensional Anderson model. |
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