Rigidity and Tolerance for Perturbed Lattic

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 50229 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Haji Mirsadeghi, Mir Omid
Abstract:
A perturbed lattice is a point process π={x+Y_x ∶x ∈Z^d} where the lattice points in Z^d are perturbed by i.i.d . random variables {Y_x }_(x∈Z^d ). A random point process π is said to be rigid if π∩B_1 (0) , the number of points in a ball , can be exactly determined given π∩B_1^c (0) , 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~N_d (0,σ^2 I) are Gaussian vectors 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 > 3 there exists a critical parameter σ_r (d) such that is rigid if σ_r (d)>σ and deletion tolerant (hence non-rigid) if σ_r (d)<σ
Keywords:
Tolerance ; Rigidity ; Lattice Theory ; Point Process ; Perturbed Lattic