Yi Sun (1) Algebraic structures for multilevel eigenvalue densities
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 13/07/2017

This talk concerns two results on the algebraic structure of multilevel densities for sample covariance matrices. In the first half of the talk, we study the static multilevel structure of certain generalized Wishart and Jacobi ensembles. In the real Wishart case, which corresponds to the spiked covariance model, we identify the process with one defined via branching of multivariate Bessel functions, generalizing a result of Dieker-Warren. In the real and complex Jacobi case, we identify the process as a principally specialized Heckman-Opdam measure, resolving a conjecture of Borodin-Gorin. In the second half of the talk, we define Laguerre and Jacobi analogues of the Warren process. That is, we construct local dynamics on a triangular array of particles so that (1) the projections to each level recover the Laguerre and Jacobi eigenvalue processes of König-O'Connell and Doumerc and (2) the fixed time distributions recover the joint distribution of eigenvalues in multilevel Laguerre and Jacobi random matrix ensembles. Our techniques extend and generalize the framework of intertwining diffusions developed by PalShkolnikov. (Columbia University) 
Presented at the 27th Annual PCMI Summer Session, Random Matrices, held June 25 – July 15, 2017. The residential, three-week Summer Session is the flagship activity of the IAS/Park City Mathematics Institute (PCMI).

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