  This work is a companion paper of Gamboa, Nagel, Rouault (J. Funct. Anal.
2016). 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 ensemble. In the gapped regime we give a probabilistic
interpretation of a Simon sum rule. The second matrix model is the Hua-Pickrell
ensemble. Unlike the Gross-Witten ensemble the potential is here infinite at
one point. Surprisingly, but as in the above mentioned paper, we obtain a
completely new sum rule for the deviation to the equilibrium measure of the
Hua-Pickrell ensemble. The extension to matrix measures is also studied.
