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  • Open Daily: 10am - 10pm
    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
612-822-4611
Global Irregularities for Poisson Processes - Gravitational Allocation and Rough Isometries.

Global Irregularities for Poisson Processes - Gravitational Allocation and Rough Isometries.

Peled, Ron

Paperback

General Mathematics

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ISBN10: 1243537981
ISBN13: 9781243537980
Publisher: Proquest Umi Dissertation Pub
Pages: 118
Weight: 0.50
Height: 0.25 Width: 7.44 Depth: 9.69
Language: English
In this dissertation we investigate two concepts which quantify global irregularities of Poisson point processes: The Gravitational Allocation and rough isometries (quasi-isometries). In the first part of the dissertation, for d >= 3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in Rd, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the allocation diameter, defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound PX>R 2, where: alphad = d-2d for d >= 4; alpha3 can be taken as any number R). In the second part of the dissertation, we investigate the question of whether two independent samples from a one-dimensional Poisson process are rough isometric almost surely. Intuitively, two metric spaces are rough isometric if their metric structure is the same in the large scale, ignoring fine details. This concept has proved fundamental in the geometric study of groups. The above question originated from Abert, Szegedy and Benjamini, and Szegedy conjectured the answer is positive. Following Benjamini, we consider a finite and quantitative version of this question which we prove is equivalent to the original question. We then make progress towards the conjecture by constructing a rough isometry which gives the first non-trivial bounds in this quantitative version. Furthermore, the rough isometry we construct is monotone and we include a discussion of monotone rough isometries, their properties and an interesting lattice structure inherent in them.

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