## Dissertations

Dissertation

#### Degree

Doctor of Philosophy

Mathematics

12-5-2011

Haiyan Cai

Stein, Lana

He, Wenjie

#### Abstract

Given a bounded region of the 2-dimensional plane, a discrete set of nodes is distributed throughout according to a Poisson point process. Given some fixed, finite, real number, two nodes are said to connect and form an edge if their mutual distance is less than this number. Let G be the graph of all such edges over the set of generated nodes and let C be any set of mutually connected nodes. It is shown that there is a critical mutual distance such that at least half of all generated nodes are mutually connected to form a connected cluster. Now, suppose that the 2-dimensional plane is partitioned into hexagons chosen such that each can be inscribed into a circle of radius which is half the size of the mutual distance. Define another notion of connectivity on the generated nodes by saying that two nodes connect if each lies in the same hexagon or each lies in a hexagon which shares a common face with the hexagon containing the other node. It is shown that the original graph of edges contains the new graph of edges and that there is an inequality relationship among the critical mutual distances. Finally, using results mentioned above, upper and lower bounds on the probability of connecting slightly more or slight less than half of all generated nodes are obtained and used to estimate the length of the interval of radii such that the probability of connecting at least half of all generated nodes will increase from some small positive value to a value near 1. This interval of connectivity radii is called a sharp threshold interval.

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