The (d+1)D solid-on-solid model is a simple model of integer-valued height functions that approximates the low-temperature interface of an Ising model. When $d\geq 2$, with zero-boundary conditions, at low temperatures the surface is localized about height 0, but when constrained to take only non-negative values entropic repulsion pushes it to take typical heights of $O(\log n)$. I will describe the mechanism of entropic repulsion and present results on how the picture changes when one introduces a competing force trying to keep the interface localized (either an external field or a reward for points where the height is exactly zero). Along the way, I will outline rich predictions for the shapes of level curves and for metastability phenomena in the Glauber dynamics.
This is based on joint works with Eyal Lubetzky and Joseph Chen.