Rigidity and Tolerance for Perturbed Lattices

Consider a perturbed lattice ${v+Y_v}$ obtained by adding IID $d$-dimensional Gaussian variables ${Y_v}$ to the lattice points in $Z^d$. Suppose that one point, say $Y_0$, is removed from this perturbed lattice; is it possible for an observer, who sees just the remaining points, to detect that a point is missing?
In one and two dimensions, the answer is positive: the two point processes (before and after $Y_0$ is removed) can be distinguished by counting points in a large ball and averaging over its radius (cf. Sodin-Tsireslon (2004) and Holroyd and Soo (2011) ). The situation in higher dimensions is more delicate, as this counting approach fails; our solution depends on a game-theoretic idea, in one direction, and on the unpredictable paths constructed by Benjamini, Pemantle and the speaker (1998), in the other.