Weakly Admissible Vector Equilibrium Problems

Abstract

We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This in particular implies the existence and uniqueness of a minimizer for such vector equilibrium problems. Our work extends earlier results in that we allow unbounded supports without having strongly confining external fields. To deal with the possible noncompactness of supports, we map the vector equilibrium problem onto the Riemann sphere and our results follow from a study of vector equilibrium problems on compacts in higher dimensions. Our results cover a number of cases that were recently considered in random matrix theory and for which the existence of a minimizer was not clearly established yet.

16 pages

References in corpus (5)

• Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
• A vector equilibrium problem for the two-matrix model in the quartic/quadratic case
• Large deviations for disordered bosons and multiple orthogonal polynomial ensembles
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Cited by in corpus (22)

• A Note on Large Deviations for 2D Coulomb Gas with Weakly Confining Potential
• Zero distribution for Angelesco Hermite--Padé polynomials
• Local incompressibility estimates for the Laughlin phase
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• Critical measures for vector energy: global structure of trajectories of quadratic differentials
• A critical phenomenon in the two-matrix model in the quartic/quadratic case
• Matrix-valued orthogonal polynomials related to hexagon tilings
• A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles
• Large Deviations for a Non-Centered Wishart Matrix
• Universality and critical behavior in the chiral two-matrix model
• Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel
• On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials
• An equilibrium problem on the sphere with two equal charges
• Large n limit for the product of two coupled random matrices
• Asymptotic analysis of the two matrix model with a quartic potential
• Sharp deviation inequalities for the 2D Coulomb gas and Quantum hall states, I
• Orthogonal polynomials in the normal matrix model with two insertions
• Vector Energy and Large Deviation
• Constrained minimum Riesz energy problems for a condenser with intersecting plates
• Logarithmic potential theory and large deviation
• Painlevé kernels in Hermitian matrix models
• Strong asymptotics for the Pollaczek multiple orthogonal polynomials ensembles