Translation invariant random harmonic functions

We will discuss the problem of existence of a translation invariant probability measure on the spaces of harmonic and discreate harmonic functions. The existence of such measures on the space of continuous harmonic functions was proved by Weiss in the late 1990s. Recently Buhovsky, Glücksam, Logunov, and Sodin found a relatively sharp lower bound for the growth of entire functions in the support of such measures. We plan to survey those results and show that there are no translation invariant probability measures on the space of discrete harmonic functions on the plane lattice.