Spectra of overlapping Wishart matrices and the Gaussian free field

Consider a doubly-infinite array of i.i.d. centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in [A. Borodin, Clt for spectra of submatrices of Wigner random matrices, Mosc. Math. J. 14(1) (2014) 29-38; A. Borodin and V. Gorin, General beta Jacobi corners process and the Gaussian free field, preprint (2013), arXiv: 1305.3627; T. Johnson and S. Pal, Cycles and eigenvalues of sequentially growing random regular graphs, Ann. Probab. 42(4) (2014) 1396-1437], can be described in terms of the height function, and leads to a connection with the Gaussian Free Field on the upper half-plane. Finally, we generalize our results from univariate polynomials to a special class of planar functions.