logarithmic potential theory
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Author(s):  
Thomas Bloom ◽  
Norman Levenberg ◽  
Vilmos Totik ◽  
Franck Wielonsky

2013 ◽  
Vol 02 (01) ◽  
pp. 1250016 ◽  
Author(s):  
ADRIEN HARDY ◽  
ARNO B. J. KUIJLAARS

We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic potential theory, which in our case is a quadratic map involving the logarithmic energies, or Voiculescu's entropies, of two measures in the presence of an external field and an upper constraint. The proof is based on a two type particles Coulomb gas representation for the eigenvalue distribution, which gives a new insight on why such variational problems should describe the limiting spectral distribution. This representation is available because of a Nikishin structure satisfied by the weights of the multiple orthogonal polynomials hidden in the background.


2012 ◽  
Vol 01 (02) ◽  
pp. 1150009 ◽  
Author(s):  
FLORENT BENAYCH-GEORGES ◽  
FRANÇOIS CHAPON

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance 1/n. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension n goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on more general Gaussian quaternionic random matrix models are also made.


2003 ◽  
Vol 10 (3) ◽  
pp. 573-593
Author(s):  
V. Maz'ya ◽  
A. Soloviev

Abstract Boundary integral equations in the logarithmic potential theory are studied by the direct method under the assumption that the contour has a peak.


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