We study existence and universality of scaling limits for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward's equation -an integro-differential identity satisfied by the rescaled one-point function.

We give a method for taking microscopic limits of normal matrix ensembles. We apply this method to study the behaviour near certain types of singular points on the boundary of the droplet. Our investigation includes ensembles without restrictions near the boundary, as well as hard edge ensembles, where the eigenvalues are confined to the droplet. We establish in both cases existence of new types of determinantal point fields, which differ from those which can appear at a regular boundary point, or in the bulk.2010 Mathematics Subject Classification. Primary: 60B20. Secondary: 60G55; 81T40; 30C40; 30D15; 35R09.

In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie derivatives. Based on this approach, we explain some equations of conformal field theory and outline their relation to SLE theory. iii iv CONTENTS 7.5. Singular vectors Appendix 8. Existence of the Virasoro field Appendix 9. Operator algebra formalism 9.1. Construction of (local) operator algebras from holomorphic Fock space fields 9.2. Radial ordering 9.3. Commutation identity and normal ordering Lecture 10. Modifications of the Gaussian free field 10.1. Construction 10.2. Vertex fields 10.3. Level two degeneracy and BPZ equations 10.4. Boundary conditions and insertions Appendix 11. Current primary fields and KZ equations 11.1. Current primary fields 11.2. KZ equations Lecture 12. Multivalued conformal Fock space fields 12.1. Chiral bosonic fields 12.2. Chiral bi-vertex fields 12.3. Rooted vertex fields Appendix 13. CFT and SLE numerology Lecture 14.

We consider various asymptotic scaling limits N → ∞ for the 2N complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials Q(ζ) = |ζ| 2λ − (2c/N ) log |ζ|, with λ > 0 and c > −1, the limiting kernel obeys a linear differential equation of fractional order 1/λ at the origin. For integer m = 1/λ it can be solved in terms of Mittag-Leffler functions. In the last part, we derive Ward's equation for planar symplectic ensembles for a general class of potentials. It serves as a tool to investigate the Gaussian and singular Mittag-Leffler universality class. This allows us to determine the functional form of all possible limiting kernels (if they exist) that are translation invariant, up to their integration domain.

In this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (“Mittag-Leffler fields”) and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for $\log |p_{n}(\zeta )|$ log | p n ( ζ ) | where pn is the characteristic polynomial of an n:th order random normal matrix.