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This paper introduces a (2+1)-dimensional Gaussian field which has theGaussian free field on the upper half-plane with zero boundary conditions ascertain two-dimensional sections. Along these sections, called space-likepaths, it matches the Gaussian field from eigenvalues of random matrices andfrom a growing random surface. However, along time-like paths the behavior isdifferent. The Gaussian field arises as the asymptotic fluctuations in quantumrandom walks on quantum groups U_q(gl_n). This quantum random walk is aq-deformation of previously considered quantum random walks. When restricted tothe space-like paths, the moments of the quantum random walk match the momentsof the growing random surface.

A (2+1)-dimensional Gaussian field as fluctuations of quantum random walks on quantum groups