Sum rules and large deviations for spectral measures on the unit circle

Abstract

This work is a companion paper of [F. Gamboa, J. Nagel and A. Rouault, Sum rules via large deviations, J. Funct. Anal. 270 (2016) 509–559] and [F. Gamboa, J. Nagel and A. Rouault, Sum rules and large deviations for spectral matrix measures, preprint (2016), arXiv:1601.08135 ] (see also [J. Breuer, B. Simon and O. Zeitouni, Large deviations and sum rules for spectral theory — A pedagogical approach, to appear in J. Spectr. Theory, preprint (2016), arXiv:1608.01467 ]). We continue to explore the connections between large deviations for random objects issued from random matrix theory and sum rules. Here, we are concerned essentially with measures on the unit circle whose support is an arc that is possibly proper. We particularly focus on two-matrix models. The first one is the Gross–Witten (GW) ensemble. In the gapped regime, we give a probabilistic interpretation of a Simon sum rule. The second matrix model is the Hua–Pickrell (HP) ensemble. Unlike the GW ensemble the potential is here infinite at one point. Surprisingly, but as in [F. Gamboa, J. Nagel and A. Rouault, Sum rules via large deviations, J. Funct. Anal. 270 (2016) 509–559], we obtain a completely new sum rule for the deviation to the equilibrium measure of the HP ensemble. The case of spectral matrix measures is also studied. Indeed, in the case of HP ensemble, we extend our earlier works on large deviation for spectral matrix measure [F. Gamboa, J. Nagel and A. Rouault, Sum rules and large deviations for spectral matrix measures, preprint (2016), arXiv:1601.08135 ] and get here also a completely new sum rule.