scholarly journals Extremal measure and external field for two parameters vector equilibrium logarithmic potential problem

2016 ◽  
pp. 1-20 ◽  
Author(s):  
Maria Alexandrovna Lapik
Keyword(s):  
External Field ◽  
Potential Problem ◽  
Logarithmic Potential ◽  
Extremal Measure ◽  
Vector Equilibrium ◽  
Two Parameters

scholarly journals LARGE DEVIATIONS FOR A NON-CENTERED WISHART MATRIX

2013 ◽  
Vol 02 (01) ◽  
pp. 1250016 ◽  
Author(s):  
ADRIEN HARDY ◽  
ARNO B. J. KUIJLAARS
Keyword(s):  
Large Deviation ◽  
Variational Problems ◽  
Logarithmic Potential ◽  
Rate Function ◽  
Coulomb Gas ◽  
Logarithmic Potential Theory ◽  
Additive Perturbation ◽  
Two Measures ◽  
Multiple Orthogonal Polynomials ◽  
Vector Equilibrium

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.

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