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