Double-scaling limits of Toeplitz determinants and their applications

For sufficiently smooth symbols, Szegö’s theorems describe the asymptotic behavior of Toeplitz determinants. 
Asymptotic expansions are also known for the determinants of Toeplitz matrices generated by Fisher-Hartwig symbols (functions that may possess zeros, integrable singularities and discontinuities). Suppose now that the symbol has an extra parameter t with the property that when t goes to zero, the number of Fisher-Hartwig singularities changes. By double-scaling limits of Toeplitz determinants we mean limits of the determinants when the size of the matrices goes to infinity and t goes to zero simultaneously. In this talk, I discuss the known results on double-scaling limits and their applications in random matrix theory.