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We study the interface of the Ising model in a box of side-length $n$ in$\mathbb Z^3$ at low temperature $1/\beta$ under Dobrushin's boundaryconditions, conditioned to stay in a half-space above height $h$ (a hardfloor). Without this conditioning, Dobrushin showed in 1972 that typically mostof the interface is flat at height $0$. With the floor, for small $h$, themodel is expected to exhibit {\it entropic repulsion}, where the typical heightof the interface lifts off of $0$. Detailed understanding of the SOS model -- amore tractable height function approximation of 3D Ising -- due to Caputo etal., suggests that there is a single integer value $-h_n^* \sim -c\log n$ ofthe floor height, delineating the transition between rigidity at height $0$ andentropic repulsion.  We identify an explicit $h_n^*=( c_\star+o(1))\log n$ such that, for thetypical Ising interface above a hard floor at $h$, all but an$\epsilon(\beta)$-fraction of the sites are propelled to be above height $0$ if$h < h_n^*-1$, whereas all but an $\epsilon(\beta)$-fraction of the sitesremain at height $0$ if $h\geq h_n^*$. Further, $c_\star$ is such that thetypical height of the unconditional maximum is $(2c_\star + o(1))\log n$; thisconfirms scaling predictions from the SOS approximation.

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