Uniform Hyperbolicity for Szego Cocycles and Applications to Random CMV Matrices and the Ising Model

作者：Damanik David^*; Fillman Jake; Lukic Milivoje; Yessen William

来源：International Mathematics Research Notices, 2015, 2015(16): 7110-7129.

DOI：10.1093/imrn/rnu158

We consider products of the matrices associated with the Szego recursion from the theory of orthogonal polynomials on the unit circle and show that under suitable assumptions, their norms grow exponentially in the number of factors. In the language of dynamical systems, this result expresses a uniform hyperbolicity statement. We present two applications of this result. On the one hand, we identify explicitly the almost sure spectrum of extended CMV matrices with nonnegative random Verblunsky coefficients. On the other hand, we show that no Ising model in one dimension exhibits a phase transition. Also, in the case of dynamically generated interaction couplings, we describe a gap labeling theorem for the Lee-Yang zeros in the thermodynamic limit.

• 出版日期2015