Asymptotics for averages over classical orthogonal ensembles

Averages of multiplicative eigenvalue statistics of Haar distributed unitary matrices are Toeplitz determinants, and asymptotics for these determinants are now well understood for large classes of symbols, including symbols with gaps and (merging) Fisher-Hartwig singularities. Similar averages for Haar distributed orthogonal matrices are Toeplitz+Hankel determinants. Some asymptotic results for these determinants are known, but not in the same generality as for Toeplitz determinants. I will explain how one can systematically deduce asymptotics for averages in the orthogonal group from those in the unitary group, using a transformation formula and asymptotics for certain orthogonal polynomials on the unit circle, and I will show that this procedure leads to asymptotic results for symbols with gaps or (merging) Fisher-Hartwig singularities. The talk will be based on joint work with Gabriel Glesner, Alexander Minakov and Meng Yang.