Rigidity and tolerance for perturbed lattices

Abstract: A perturbed lattice is a point process $\Pi={x+Y_x:x\in \mathbb{Z}^d}$ where the lattice points in $\mathbb{Z}^d$ are perturbed by i.i.d.\ random variables ${Y_x}_{x\in \mathbb{Z}^d}$. A random point process $\Pi$ is said to be rigid if $|\Pi\cap B_0(1)|$, the number of points in a ball, can be exactly determined given $\Pi \setminus B_0(1)$, the points outside the ball. The process $\Pi$ is called deletion tolerant if removing one point of $\Pi$ yields a process with distribution indistinguishable from that of $\Pi$. Suppose that $Y_x\sim N_d(0,\sigma^2 I)$ are Gaussian vectors with with $d$ independent components of variance $\sigma^2$. Holroyd and Soo showed that in dimensions $d=1,2$ the resulting Gaussian perturbed lattice $\Pi$ is rigid and deletion intolerant. We show that in dimension $d\geq 3$ there exists a critical parameter $\sigma_r(d)$ such that $\Pi$ is rigid if $\sigma<\sigma_r$ and deletion tolerant (hence non-rigid) if $\sigma>\sigma_r$.