Rigidity and tolerance for perturbed lattices

Peres, Y., Microsoft Research

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? Holroyd and Soo (2011) noted that in one and two dimensions, the answer is positive: the two point processes (before and after Y_0 is removed) can be distinguished using smooth statistics, analogously to work of Sodin and Tsirelson (2004) on zeros of Gaussian analytic functions. The situation in higher dimensions is more delicate, with a phase transition that 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. I will also describe a related point process where removal of one point can be detected but not the removal of two points. (Joint work with Allan Sly, UC Berkeley).