Large Deviations for a Non-Centered Wishart Matrix
arXiv:1204.6261 · doi:10.1142/S2010326312500165
Abstract
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.
40 pages
References in corpus (9)
- Biorthogonal ensembles
- Orthogonal polynomial ensembles in probability theory
- A note on biorthogonal ensembles
- On some integrals over the U(N) unitary group and their large N limit
- Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
- Weakly Admissible Vector Equilibrium Problems
- A Note on Large Deviations for 2D Coulomb Gas with Weakly Confining Potential
- A vector equilibrium problem for the two-matrix model in the quartic/quadratic case
- Character expansion method for the first order asymptotics of a matrix integral