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The thin obstacle problem is a free boundary problem concerning the shape of an elastic membrane resting on a lower-dimensional obstacle. In this talk, we discuss some rigidity properties of solutions in the entire space. We see very rigid behaviors when the solution grows quadratically at infinity. When the solution has higher rate of growth, we see no rigidity. This talk is based on joint works with Simon Eberle (BCAM) and Xavier Fernandez-Real (EPFL).