Asymptotic study of Toeplitz determinants with Fisher-Hartwig symbols and their double-scaling limits

Alahmadi, R. (2024) Asymptotic study of Toeplitz determinants with Fisher-Hartwig symbols and their double-scaling limits. PhD thesis, University of Reading. doi: 10.48683/1926.00117924

Abstract/Summary

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.

Item Type	Thesis (PhD)
URI	https://reading-clone.eprints-hosting.org/id/eprint/117924
Identification Number/DOI	10.48683/1926.00117924
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
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