It is also shown that when restricting to the Gelfand–Tsetlin subalgebra of U(glN), the non-commutative Markov operator matches a (2+1)-dimensional random surface model introduced in [7]. As an application, it is then proved that the moments converge to an explicit Gaussian field along time-like paths. Combining with [7] which showed convergence to the Gaussian free field along space-like paths, this computes the entire three-dimensional Gaussian field. In particular, it matches a Gaussian field from eigenvalues of random matrices [5].