  We consider a continuous time random walk on the two-dimensional discrete
torus, whose motion is governed by the discrete Gaussian free field on the
corresponding box acting as a potential. More precisely, at any vertex the walk
waits an exponentially distributed time with mean given by the exponential of
the field and then jumps to one of its neighbors, chosen uniformly at random.
We prove that throughout the low-temperature regime and at in-equilibrium
timescales, the process admits a scaling limit as a spatial K-process driven by
a random trapping landscape, which is explicitly related to the limiting
extremal process of the field. Alternatively, the limiting process is a
supercritical Liouville Brownian motion with respect to the continuum Gaussian
free field on the box. This demonstrates rigorously and for the first time, as
far as we know, a dynamical freezing in a spin glass system with
logarithmically correlated energy levels.
