Construction Of New Family Research Articles

Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter

A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.

A Construction of New Families of Minimal Lagrangian Submanifolds via Torus Actions

In this paper we investigate connections between minimal Lagrangian submanifolds and holomorphic vector fields in Kahler manifolds. Our main result is: Let M2n (n ≥ 2) be a Kahler-Einstein manifold with positive scalar curvature with an effective, structure-preserving action by an n-torus Tn. Then precisely one regular orbit L of the Tn-action is a minimal Lagrangian submanifold of M. Moreover there is an (n − 1)-torus Tn−1 ⊂ Tn and a sequence of non-flat immersed minimal Lagrangian tori Lk in M such that all Lk are invariant under Tn−1 and Lk locally converge to L (in particular the supremum of the sectional curvatures of Lk and the distance between L and Lk go to 0 as k ↦ ∞). This result is new even for M = ℂPn for n ≥ 3.