Hard Edge Statistics of Products of P\'olya Ensembles and Shifted GUE's

Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative P\'olya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes transform (Green function). Under rather mild conditions for the P\'olya ensemble, we prove formulas for the hard edge kernel of the singular value statistics of the P\'olya ensemble alone and the product matrix to highlight their very close similarity. Due to these observations, we even propose a conjecture for the hard edge statistics of a multiplicative P\'olya ensemble on the complex matrices and a polynomial ensemble on the Hermitian matrices.