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

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