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 $d\ge 3$ there exists a critical parameter ${\sigma }_r(d)$ such that $\Pi$ is rigid if $\sigma <{\sigma }_r$ and deletion tolerant (hence non-rigid) if $\sigma >{\sigma }_r$.

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