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Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with absolutely continuous spectrum

Artur Avila, Yoram Last and Barry Simon

Vol. 3 (2010), No. 1, 81–108

By combining ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for orthogonal polynomials on the real line in the absolutely continuous spectral region is implied by convergence of 1 nKn(x,x) for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with absolutely continuous spectrum and prove that the limit of 1 nKn(x,x) is ρ(x)w(x), where ρ is the density of zeros and w is the absolutely continuous weight of the spectral measure.

orthogonal polynomials, clock behavior, almost Mathieu equation
Mathematical Subject Classification 2000
Primary: 26C10, 42C05, 47B36
Received: 20 October 2009
Accepted: 19 November 2009
Published: 4 March 2010
Artur Avila
Laboratoire de Probabilités et Modèles Aléatoires
Université Pierre et Marie Curie
Boîte Courrier 188
75252 Paris Cedex 05
Instituto Nacional de Matemática Pura e Aplicada
Estrada Dona Castorina 110
22460-320 Rio de Janeiro, RJ
Yoram Last
Institute of Mathematics
The Hebrew University
91904 Jerusalem
Barry Simon
Department of Mathematics
California Institute of Technology
MC 253-37
Pasadena, CA 91125
United States