Rigidity and tolerance for perturbed lattices

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

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