Asymptotics of Toeplitz determinants and quantum spin chain models

Jani Virtanan, Reading University. Part of the analysis and applications seminar series.

This talk is concerned with asymptotic analysis of the determinants of Toeplitz matrices (defined as matrices constant along the parallels to the main diagonal) as their size goes to infinity.

The entries of a Toeplitz matrix are given by the Fourier coefficients of an integrable function on the unit circle, which we call the symbol of the Toeplitz matrix. For symbols that are sufficiently smooth or possess Fisher-Hartwig singularities, the asymptotic behavior of Toeplitz determinants is well understood. If the symbol has an extra parameter, it is of considerable interest to compute the double-scaling limits of Toeplitz determinants as their size goes to infinity and the parameter goes to some critical value simultaneously.

Recent results on double-scaling limits and their applications in random matrix theory and the theory of quantum systems will be discussed.