On the One Dimensional Poisson Random Geometric Graph

On the One Dimensional Poisson Random Geometric Graph

Topological properties of the one dimensional exponential random geometric graph

Connectivity in One-Dimensional Geometric Random Graphs: Poisson Approximations, Zero- One Laws and Phase Transitions

Consider n points (or nodes) distributed uniformly and independently on the unit interval [0, 1]. Two nodes are said to be adjacent if their distance is less than some given threshold value. For the underlying random graph we derive zero-one laws for the property of graph connectivity and give the asymptotics of the transition widths for the associated phase transition. These results all flow f...

On the bounds in Poisson approximation for independent geometric distributed random variables

‎The main purpose of this note is to establish some bounds in Poisson approximation for row-wise arrays of independent geometric distributed random variables using the operator method‎. ‎Some results related to random sums of independent geometric distributed random variables are also investigated.

One-Dimensional Poisson Growth Models With Random and Asymmetric Growth

Suppose there is a Poisson process of points Xi on the line. Starting at time zero, a grain begins to grow from each point Xi, growing at rate Ai to the left and rate Bi to the right, with the pairs (Ai, Bi) being iid. A grain stops growing as soon as it touches another grain. When all growth stops, the line consists of covered intervals (made up of contiguous grains) separated by gaps. We show...