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Riemann-Hilbert Problems and their applications in mathematical physics

Kozlowska, K. (2017) Riemann-Hilbert Problems and their applications in mathematical physics. PhD thesis, University of Reading

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The aim of this thesis is to present the reader with the very effective and rigorous Riemann-Hilbert approach of solving asymptotic problems. We consider a transition problem for a Toeplitz determinant; its symbol depends on an additional parameter t. When t > 0, the symbol has one Fisher-Hartwig singularity at an arbitrary point z1 6= 1 on the unit circle (with associated α1, β1 ∈ C strengths) and as t → 0, a new Fisher-Hartwig singularity emerges at the point z0 = 1 (with α0, β0 ∈ C strengths). The asymptotics we present for the determinant are uniform for sufficiently small t. The location of the β-parameters leads to the consideration of two cases, both of which are addressed in this thesis. In the first case, when | Re β0 − Re β1| < 1 we see a transition between two asymptotic regimes, both given by the same result by Ehrhardt, but with different parameters, thus producing different asymptotics. In the second case, when | Re β0 − Re β1| = 1 the symbol has Fisher-Hartwig representations at t = 0, and the asymptotics are given the Tracy-Basor conjecture. These double scaling limits are used to explain transition in the theory of XY spin chains between different regions in the phase diagram across critical lines.

Item Type:Thesis (PhD)
Thesis Supervisor:Virtanen, J. and Pelloni, B.
Thesis/Report Department:Department of Mathematics and Statistics
Identification Number/DOI:
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:73488

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