Speaker
Duits, KTH Royal Institute of Technology
Abstract
By virtue of the Strong Szegő Limit Theorem for Toeplitz determinants,
linear statistics of the CUE with sufficiently smooth test functions satisfy
a central limit theorem with no \(n\)-dependent normalization. In particular,
the variance converges and the cumulants of order at least three tend to zero.
When the test function has jump and/or logarithmic singularities,
Fisher–Hartwig asymptotics imply that the variance grows logarithmically
with \(n\), while cumulants of order at least three remain bounded. This
behavior has played an important role in recent proofs of convergence to
Gaussian multiplicative chaos measures associated with the characteristic
polynomial of the CUE.
In this talk, I will report on joint work with Nedialko Bradinoff, in which
we show that the same behavior holds for a larger class of test functions
whose Fourier coefficients decay slowly. In other words, the structure of
the Fisher–Hartwig asymptotics is not a consequence of the singularities
themselves, but rather of the presence of high frequencies. The class we
study includes continuous test functions, as well as functions with dense
sets of logarithmic singularities.