Integrable Systems and Random Matrix Theory Seminar

Among the numerous appearances of Toeplitz determinants in Mathematics, Physics, and Engineering problems, one of the most outstanding is the groundbreaking discovery of Kaufman and Onsager who established that the diagonal and horizontal two-point correlation functions in the square lattice Ising model have Toeplitz determinant representations. In another interesting development in 1987, Au-Yang and Perk expressed the next-to-diagonal correlations of the anisotropic Ising model in terms of a bordered Toeplitz determinant: a determinant with Toeplitz structure except for its last row or column.

In my talk, after motivating the problem I will explain how a class of bordered Toeplitz determinants is encoded in the solution of the Baik-Deift-Johansson Riemann-Hilbert problem for biorthogonal polynomials on the unit circle. This approach is inspired by the 2007 work of Witte in which the connection of certain bordered Toeplitz determinants to the system of biorthogonal polynomials on the unit circle was found. Finally, I will show that by applying our general results to the Ising case, we can rigorously confirm the already heuristically known result that in the low-temperature regime, the next-to-diagonal correlations are the same as the diagonal and horizontal ones.

This talk is based on joint work with Estelle Basor, Torsten Ehrhardt, Alexander Its, and Yuqi Li, where we independently employ Riemann-Hilbert, operator-theoretic and numerical methods to obtain the asymptotics of a class of bordered Toeplitz determinants as the size of the matrix tends to infinity. Speaker(s): Roozbeh Gharakhloo (Colorado State University)