Three-dimensional Gaussian fluctuations of non-commutative random surfaces along time-like paths.

We construct a continuous-time non-commutative Markov operator on U ( gl N ) with corresponding morphisms U ( gl N ) → End ( L 2 ( U ( N ) ) ) ⊗ ∞ at any fixed times 0 ≤ t 1 < t 2 < … . This is an analog of a continuous-time non-commutative Markov operator on the group von Neumann algebra v N ( U ( N ) ) constructed in [16] , and is a variant of discrete-time non-commutative random walks on U ( gl N ) [2,9] . It is also shown that when restricting to the Gelfand–Tsetlin subalgebra of U ( gl N ) , 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] . [ABSTRACT FROM AUTHOR]