Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle

2004/11/17 by Barry Simon, Simon, Barry
Mathematics · #26C05 #47N20 #82B44 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:26C05 #msc:47N20 #msc:82B44

paper · pdf · doi:10.48550/arxiv.math/0411388

Keywords: OPUC, random Verblunsky coefficients, localization

arxiv created 2004/11/17 · arxiv updated 2009/12/01

Abstract

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if 𝔼 \biggl(∫(dθ)/(2π) \biggl|\biggl(\fracC + e^iθC -e^iθ \biggr)_kℓ\biggr|^p \biggr) ≤ C₁ e^{-κ₁ |k-ℓ|} for some κ₁ >0 and p<1, then for suitable C₂ and κ₂ >0, 𝔼 (sup_n |(Cⁿ)_kℓ|) ≤ C₂ e^{-κ₂ |k-ℓ|} Here C is the CMV matrix.

Related

• Orthogonal Polynomials, Verblunsky Coefficients, and a Szegő-Verblunsky Theorem on the Unit Sphere in ℂ^d
• Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle
• Coefficients of Orthogonal Polynomials on the Unit Circle and Higher Order Szego Theorems
• Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift
• A CMV connection between orthogonal polynomials on the unit circle and\n the real line