Asymptomatic study of Toeplitz determinants and Fisher-Hartwig symbol and their double-scaling limits

Abstract

This thesis aims to study the asymptotic behavior of Toeplitz determinants Dn(ft(z)) by using the Riemann-Hilbert analysis. We consider the double scaling limits of Toeplitz determinants with respect to symbol ft(z). This symbol possess m Fisher-Hartwig singularities when t > 0, and m + 1 if t → 0. We obtain the uniform asymptotics for Dn(ft(z)) as n → ∞ which is valid for all sufficiently small t in terms of Painlev´e V function. This study is divided into two parts: We first consider the case when the seminorm |||β (t) ||| < 1 for t ≥ 0 and then the case of the Basor-Tracy asymptotics when |||β (t) ||| = 1 for some t. The latter case is further divided to the cases, |||β (t) ||| < 1 for t > 0 and |||β (t) ||| = 1 for t > 0. In the last chapter we present the computation of the magnetization of the 2D Ising model in the high temperature regime T > Tc (i.e., t < 0) including all the details by using the Riemann-Hilbert approach and the asymptotics of Toeplitz determinants.