We study fluctuations of polynomial linear statistics for discrete Schrödinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuations.
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We are grateful to Ofer Zeitouni for a useful discussion, and to the anonymous referee for useful comments. Research by JB and YG was supported in part by the Israel Science Foundation (Grant No. 399/16) and in part by the United States-Israel Binational Science Foundation (Grant No. 2014337). Research by MW was supported in part by the Israel Science Foundation (Grant No. 1612/17) and in part by the United States-Israel Binational Science Foundation (Grant No. 2006066).
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