Riemann-Hilbert methods in general relativity and random matrix theory

Sammanfattning: This thesis focuses on problems in general relativity and random matrix theory that can be solved by means of Riemann-Hilbert techniques. The first part of the thesis is dedicated to solving boundary value problems related to colliding plane waves in vacuum in general relativity. As the relevant partial differential equations are integrable, the boundary value problems can be related to Riemann-Hilbert problems via the so-called inverse scattering transform. The second part of the thesis is concerned with large gap asymptotic expansions for certain determinantal point processes. The considered large gap probabilities can be related to Riemann-Hilbert problems via a procedure by Its, Izergin, Korepin and Slavnov (IIKS) and hence asymptotics can be studied by applying the so-called nonlinear steepest descent method.The thesis consists of five papers as well as a background chapter and a chapter summarizing the results.In Paper A we study a Goursat problem for the Euler-Darboux equation. This boundary value problem describes the collision of two colliding gravitational plane waves with collinear polarization. In the paper we use classical techniques to compute an asymptotic expansion to all orders of the solution of the Goursat problem close to the critical singular line. It is known that curvature singularities can appear at this line. We don't apply Riemann-Hilbert techniques in this paper but it can be seen in line with the Papers B and C.In Paper B we study a Goursat problem for the Ernst equation which describes the collision of two gravitational plane waves which are not necessarily collinearly polarized. The main result of the paper is a new representation formula for the general solution of the Goursat problem in terms of a 22 Riemann-Hilbert problem. Moreover, we answer certain uniqueness and existence questions and compute the boundary behavior of the solution. The latter is important in the context of colliding plane waves.In Paper C we extend the results of Paper B to the situation of colliding electromagnetic plane waves in Einstein-Maxwell theory. A solution representation in terms of a 33 Riemann-Hilbert problem is derived and we extend the results of Paper B regarding existence, uniqueness and boundary behavior.In Paper D we study large gap probabilities in the hard edge scaling limit of Muttalib-Borodin ensembles with Laguerre weight. These ensembles appear for instance as eigenvalue distributions for certain random matrix models. We apply the IIKS procedure and the nonlinear steepest descent method in order to compute new constants in the large gap asymptotic expansion at the hard edge and, in particular, the multiplicative constant term. The results in this paper are based on recent results by Claeys, Girotti and Stivigny.In Paper E we compute the multiplicative constant in the large gap asymptotics for the Meijer-G point process using similar techniques as in Paper D. The Meijer-G point process appears for instance in hard edge scaling limits of certain product random matrix ensembles.