2014/09/16 by Yuval Peres, Peres, Yuval, Allan Sly +1 · 3 citations
Computer Science · Mathematics · #60K35 #Bayesian Methods and Mixture Models #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1409.4490
openalex publication_date 2014/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A perturbed lattice is a point process Π=\x+Yx:x∈ ℤd\ where the lattice points in ℤd are perturbed by i.i.d. random variables \Yx\x∈ ℤd. A random point process Π is said to be rigid if |Π∩ B0(1)|, the number of points in a ball, can be exactly determined given Π∖ B0(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 Yx∼ Nd(0,σ2 I) are Gaussian vectors with with d independent components of variance σ2. 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≥ 3 there exists a critical parameter σr(d) such that Π is rigid if σσr.