THE SITUATION:
Begin by taking a periodic strip of length 2*Pi. Then use a uniform distribution to place K points (x1,…, xk) on the strip by assigning each of them a randomly sampled number. Then draw epsilon neighborhoods around each point xi. We call two points connected if they are contained in the same epsilon neighborhood, that is, if the distance between them is less than epsilon. Note that because the strip is periodic, the distance between two points must be calculated as the minimum of the absolute distance between them and 2*Pi minus the absolute distance between them. I.e. the distance between 0.1 and 2*Pi is 0.1, but not 2*Pi – 1.
QUESTION 1: What is the function that tells us the number of connected components composed of the points (x1,…, xk) in terms of the size of the neighborhoods epsilon and the number of points K?
QUESTION 2: What is the average critical value of epsilon required in order for there to be one connected component for some value of K? This could also be answered in the form of a Probability Density Function in terms of epsilon.
