Universality at the edge for unitary matrix models (Q2836982)

Unitary matrix models are studied. These models are defined by the probability law NEWLINE\[NEWLINEp_n(U)d \mu_n(U) = Z^{-1}_{n,2}\text{exp}\left \{-n\text{Tr}V \left(\frac{U + U^*}{2} \right)\right \} d\mu_n(U),NEWLINE\]NEWLINE where \(U = \{ U_{jk}\}^n_{j,k=1}\) is an \(n \times n\) unitary matrix, \(\mu_n(U)\) is the Haar measure on the group \(U(n)\), \(Z_{n,2}\) is a normalization constant, and \(V : [-1, 1] \rightarrow {\mathbb R}\) is a continuous function called the potential of the model. As the main result, the universality conjecture is proved concerning the local eigenvalue statistics for unitary matrix models with a smooth potential \(V\) having four bounded derivatives in the case of one-interval support \(\sigma\) of the limiting normalized counting measure. This result is obtained using the orthogonal polynomial technique when a system of polynomials orthogonal on the unit circle with a varying weight is considered.