Jani Virtanen: Transition asymptotics of Toeplitz determinants and their applications in random matrix theory

The study of the asymptotics of Toeplitz determinants is
important because of a vast number of applications in random matrix
theory and mathematical physics. These asymptotics are well understood
for many symbol classes, such as smooth Szegö symbols, and symbols
with Fisher-Hartwig (F-H) singularities of jump and/or root type. By
introducing an additional parameter in the symbol, we can consider
what is called transition asymptotics. In the paper "Emergence of a
singularity for Toeplitz determinants and Painleve V," Clayes, Its and
Krasovsky considered the transition case between a Szego and a F-H
symbol with one singularity. In this talk we discuss a transition in
which we see emergence of additional singularities. In that case
however, we need to consider so-called F-H representations and the
Tracy-Basor conjecture. These types of results model phase transitions
in numerous problems arising in statistical mechanics, one of which
will be mentioned in the talk. Joint work with Kasia Kozlowska.