The focus of this project is to investigate the asymptotics of determinants of perturbations of convolution operators. Our goal will be to extend the classical limit theorems to these operators, both for scalar and matrix-valued symbols, and for both smooth and singular symbols. For many of these operators, the constant term is the most difficult piece of the asymptotic expansion to describe. For matrix-valued symbols there are only a few cases where the constants can be explicitly described. In particular, we will investigate the asymptotics in the case of a perturbation of a Toeplitz determinant by a Hankel operator with possibly different symbol. Other classes of operators of interest are Wiener-Hopf plus Hankel operators and Bessel operators. Classical operator methods will be used to study these problems as well as newer developments. For example, using the BorodinGeronimo-Geronimo-Case identity to bridge between smooth and singular symbols has been highly successful.
There is increasing interest in finding asymptotic expansions of determinants of convolution type operators because they have connections to many problems in mathematical physics, including the Ising model (a model of a two-dimensional (or very thin) magnets), the classical dimer model, the entanglement problem in spin chain model, random growth models, and to the general area of random matrix theory. In these physical problems one is often interested in the complicated, unpredictable behavior of the models. Often a quantity that describes some statistical property of a system can be reformulated as a determinant approximation problem. The physical systems give predictions as to the right form of the approximation and show that many of the answers should be quite universal. The universality is especially important since it shows that many complicated systems and models are actually quite similar. Hence the idea is not simply to prove theorems and then find applications for the theorems, but to use the ideas of mathematical physics to give predictions of the mathematics and then conversely, to use the mathematics to tell us something about physical systems.
