Connecting N points pairwise in volume V, average density of lines?

[Spinnor]

Say we take N random points in a volume V and connect the points pairwise with line-segments. I would like to estimate the number of segments that intersect some small volume v, and where N is large enough so that any small random sample volume v will have many intersections. Little volume v may or may not enclose any points.

Does this get me close? First let us estimate the total length of the line segments, Ʃ. Use an average separation distance D between each pair of points. The distance D is of order one half the length of the volume V, V = L^3, D = L/2.

The total number of segments is N(N-1)/2 so an estimate for the length of line-segments,

Ʃ = D*N(N-1)/2 for large N this is about D*N^2/2
Ʃ ≈ D*N^2/2

Assume this total length is evenly divided into each small volume v. The length in volume v is the fraction [d^3/D^3] times Ʃ,

[d^3/D^3]*Ʃ = [d^3/D^3]*D*N^2/2 = σ

Assume the average length of the line-segments that intersect the little volume v is one-half the length the little volume v, d/2.

Then the average number of line-segments in v, ω, is,

ω = σ/[d/2] = {[d^3/D^3]*D*N^2/2}/[d/2] = d^2*N^2/D^2

using Wolfram calculator,

http://www.wolframalpha.com/

Using our Universe as an example, let d = 1m, N = 10^80, D = [3.5*10^80m^3]^.3333 ≈ 7*10^26

ω = 2*10^106 segments intersecting a volume of 1m^3.

We can ask what must the size of the volume v above be so that on average there will be only one line-segment intersecting it.

Set ω = 1 = d^2*N^2/D^2 now d is unknown and we use N and D above,

d = D/N = 7*10^26/10^80 = 7*10^-54m. If we are too near a point this estimate is bad. If we enclose a point ω jumps by about N

We can also ask how many points must a volume v have so that ω above changes significantly because of the additional line-segments from the enclosed points.

I made many bad estimates but I think I'm within a factor of a billion above?

Thanks for any help!

 

[Tinyboss]

This is an interesting problem! I wonder, are the endpoints important, or do they just serve as a way of defining some random lines? Would it serve your purpose equally well to have some number of lines (the infinite kind) randomly distributed throughout your space?

 

[Spinnor]

This is an interesting problem! I wonder, are the endpoints important, or do they just serve as a way of defining some random lines? Would it serve your purpose equally well to have some number of lines (the infinite kind) randomly distributed throughout your space?

I had something in mind for the endpoints. If you have N charged particles in some volume then the electrstatic potential energy is given by the sum of the potential energy of each pair of charged particles. And for N particles with mass the gravitational potential energy (Newtonian Gravity) is given by a similar pairwise sum. In either case the length of the line segments is important. So if there were some unseen connection between each pair of particles in our universe I was curious just how dense would those connections be. Just an idea. It seems for the numbers I used if there were some type of unseen connections they would have to be very dense. I also had the idea that disturbances of these connections might be gravitons and photons. Spinors would have to pop out of such an idea and I don't know if that is possible.

After some thought I realized that for N random points in a box that the density of line segments would fall off near the inside boundary of the box. Things could be more uniform in some closed space like S^3 but then a pair of points in S^3 gives two (or more) geodesic paths between the points? Maybe you include both paths?

So do you think I'm in the ballpark with those numbers?