Asymptotics of Muttalib-Borodin determinants with Fisher-Hartwig singularities

Muttalib-Borodin determinants are generalizations of Hankel determinants and depend on a parameter $\theta>0$. In this paper, we obtain large $n$ asymptotics for $n \times n$ Muttalib-Borodin determinants whose weight possesses an arbitrary number of Fisher-Hartwig singularities. As a corollary, we obtain asymptotics for the expectation and variance of the real and imaginary parts of the logarithm of the characteristic polynomial, several central limit theorems, and some global bulk rigidity upper bounds. Our results are valid for all $\theta > 0$.