We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green’s function for the associated Jacobi matrices.As a particular consequencewe obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.
Bibliographical noteFunding Information:
Maurice Duits: Supported in part by the Grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation and by the Swedish Research Council (VR) Grant No. 2012-3128.
Jonathan Breuer: Supported in part by the US-Israel Binational Science Foundation (BSF) Grant No. 2010348 and by the Israel Science Foundation (ISF) Grant No. 1105/10.
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