A Zabrodin
Normal random matrices and problems of complex analysis
Abstract:
The model of normal random matrices at large N is shown to provide
a constructive proof of recent new results on the Dirichlet boundary problem
and inverse potential problem in two dimensions. The key object of our analysis is the support of eigenvalues of the normal matrices in the planar large N limit. It is a domain D (or several disconnected domains) in the complex plane.
The correlation functions of the model are expressed through the Green
function
of the domain complementary to the D. The 1/N expansion of the free energy
of the model will be also discussed. Using the loop equation,
the next-to-leading correction to the free energy (`genus 1 contribution')
is related to the determinant of the Laplace operator in D.
lecture:
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