# Diffusive 1:2:3 scaling and low temperature interfaces

Abstract: A paradigm for the diffusive 1:2:3 scaling is a random walk above a hard wall and subject to the area tilt which scales like the reciprocal of the linear size of the system. A related phenomenon is that of critical prewetting in the 2D Ising model below critical temperature. In both cases the scaling limits are Ferrari-Spohn diffusions.

In the context of 2+1 low temperature effective interfaces, such as models of facet formation or models of entropic repulsion of Solid-On-Solid random surfaces above a hard wall, there is a natural action of area tilts on macroscopic level lines. The strength of these tilts is compatible with the 1:2:3 diffusive scaling (reciprocal of the linear size) and it may be the same for all the ordered level lines (models of facet formation) or it may depend on the serial numbers of particular level lines (entropic repulsion). Furthermore, the number of macroscopic level lines may or may not grow with the linear size of the system. There are several scaling regimes, and the corresponding questions make sense and pose a challenge even in the simplified context of non-intersecting Brownian polymers. There are some results and quite a few open problems, both of which I shall try to outline.

Based on joint works with Pietro Caputo, Sebastien Ott, Senya Shlosman, Yvan Velenik and Vitali Wachtel.