Interpolation of Gaussian Free Fields via Random Matrices

In this paper we construct a two parameter family of generalized Gaussian fields which interpolates between the Gaussian Free Field, conditioned Gaussian Free Fields, and Gaussian Free Fields with independent noise. Depending on the values of the parameters, we characterize these generalized Gaussian fields as partial conditionings of a Gaussian Free Field with perturbations of its first two Fourier modes. We show that this family arises as the fluctuations of the height function associated with the corner process of a two parameter variant of Wigner random matrices.