# Rigidity and tolerance for perturbed lattices

# 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 $\mathbb{Z}^d$.

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 using smooth

statistics, analogously to work of Sodin and Tsirelson (2004) on zeros of

Gaussian analytic functions (cf. Holroyd and Soo (2011)). The situation

in higher dimensions is more delicate; 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.

(Joint work with Allan Sly).