## Riemann-Hilbert Problems and their applications in mathematical physics
Kozlowska, K.
(2017)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. ## Abstract/SummaryThe 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.
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