Abstract
We study spacing distribution for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward’s (or the “rescaled loop”) equation—an identity satisfied by all sequential limits of the rescaled one-point functions.
Highlights
Introduction and Results1.1 Potential Theory and DropletsFix a suitable function (“external potential”) Q : C → R ∪ {+∞}
8 Concluding Remarks In Subsection 8.1, we explain how the boundary kernel K = G F in the Ginibre case can be related to asymptotics of section of the exponential function
In Subsection 8.2, we will mention some connections to the theory of Hilbert spaces of entire functions and to the theories of certain special functions
Summary
Fix a suitable function (“external potential”) Q : C → R ∪ {+∞}. Let P denote the class of positive, compactly supported Borel measures on C. We always assume that Q is lower semi-continuous and that Q is finite on some set of positive logarithmic capacity. With these assumptions, the complement Sc has a local Schwarz function at each boundary point, and we can rely on the fundamental theorem of Sakai [33] concerning domains with local Schwarz functions. We can apply Sakai’s regularity theorem, which implies that all but finitely many boundary points p ∈ ∂ S are regular in the sense that there is a disc D = D( p; ) such that D \ S is a Jordan domain and D ∩ (∂ S) is a real analytic arc. Such points can be classified further as cusps or double points
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