Global fluctuations for Multiple Orthogonal Polynomial Ensembles

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Ensembles form an important class determinantal point processes that include random matrix models such as GUE external source, complex Wishart matrices, multi-matrix and others. Our analysis is based on recurrence multiple orthogonal polynomials, constructed out nearest neighbor recurrences. If coefficients recurrences have limits, then we show right-limit this a can be viewed representation Toeplitz operator respect to non-standard basis. This will allow us prove Central Limit Theorems A particular novelty use Baker–Campbell–Hausdorff formula higher cumulants converge zero. illustrate main results by discussing Gaussian Unitary matrices specializations Schur measure related Charlier, Krawtchouk Meixner polynomials.