Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor

We study the interface of the Ising model in a box of side-length n in Z³ at low temperature 1/β under Dobrushin’s boundary conditions, conditioned to stay in a half-space above height -h (a hard floor). Without this conditioning, Dobrushin showed in 1972 that typically most of the interface is flat at height 0. With the floor, for small h, the model is expected to exhibit entropic repulsion, where the typical height of the interface lifts off of 0. Detailed understanding of the SOS model—a more tractable height function approximation of 3D Ising—due to Caputo et al., suggests that there is a single integer value -h*_n ~ -c log n of the floor height, delineating the transition between rigidity at height 0 and entropic repulsion. We identify an explicit h*_n = (c_*+о(1)) log n such that, for the typical Ising interface above a hard floor at h, all but an ε(β)-fraction of the sites are propelled to be above height 0 if h < h*_n- 1, whereas all but an ε(β)-fraction of the sites remain at height 0 if h ≥ h*_n. Further, c_* is such that the typical height of the unconditional maximum is (2c_* + о(1)) log n; this confirms scaling predictions from the SOS approximation.