We will discuss the probability measures on the space of harmonic functions on $\mathbb{R}^2$, which are invariant with respect to translations, and are not supported on constant functions. It is a wild object introduced by B. Weiss, who showed existence and even abundance of such measures. For discrete harmonic functions on the two-dimensional square lattice it appears that there are no such measures. We will discuss growth properties of harmonic functions related to existence/non-existence of this wild object.

Based on the joint works with Lev Buhovsky, Adi Glucksam, Eugenia Malinnikova and Mikhail Sodin, arXiv:1703.08101 arXiv:1712.07902