Use this URL to cite or link to this record in EThOS:  http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.729353 
Title:  RiemannHilbert problems and their applications in mathematical physics  
Author:  Kozlowska, Katarzyna 
ISNI:
0000 0004 6494 2240


Awarding Body:  University of Reading  
Current Institution:  University of Reading  
Date of Award:  2017  
Availability of Full Text: 


Abstract:  
The aim of this thesis is to present the reader with the very effective and rigorous RiemannHilbert 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 FisherHartwig singularity at an arbitrary point z1 6= 1 on the unit circle (with associated α1, β1 ∈ C strengths) and as t → 0, a new FisherHartwig 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 βparameters 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 FisherHartwig representations at t = 0, and the asymptotics are given the TracyBasor 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.


Supervisor:  Not available  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.729353  DOI:  Not available  
Share: 