The number of formal proofs in "statistical geometry" is small at present. The material here aims to provide an introduction to mathematicians who might want to improve the understanding of it.
The best place to start is with the 2013 Shier-Bourke paper. The key ideas which need to be grasped are (a) the area rule with its parameters c and N, (b) the nonoverlapping random search algorithm and the definitions of "trial" and "placement", (c) the idea of a halting probability. The last version to go to the editor (99.9% the same as the published version) can be viewed and downloaded here paper
When familiarity with the algorithm has been gained it is suggested that the serious mathematician find a proof of nonhalting for the 1-dimensional case. This is not especially difficult. John and several other people have independently found such proofs, which show that the 1D algorithm is unconditionally nonhalting when 1 < c < 2. One can find a bound for sums of terms 1/(i+N)^c (i = 0, 1, 2, ...) using the corresponding integral.
The statistical geometry algorith runs without halting and is space-filling (in the limit) for any shape or sequence of shapes which obey the area rule, when 1 < c < c_max and N > N_min.
The values of c_max and N_min are related. There must exist a c_max because as c increases it becomes impossible to place shape 1. There must exist an N_min because as N falls toward 0 the 0th area becomes completely dominant. The details depend on the particular shape of interest.
It is found that c_max depends on the shape. It has its highest value (~1.45-1.5) for circles and squares. Sparse, noncompact shapes have low c_max.
The conjecture stated above is the "maximum strength" version and is very broad. John has done 1e4s of computer runs with an immense variety of 2D shapes without finding any exceptions, and a couple of other people (e.g., Paul Bourke) have also done a lot of work. Halting studies of the 2D case have tended to begin with simple examples such as fractalization of circles within a circle, assuming N = 1.
A formal proof of nonhalting for the particular case of circles fractalized within a circle has been found by Christopher Ennis. See Math Horizons, Feb. 2016 issue, p. 8.
The links below lead to some numerical studies that while they are not proofs offer some insight for those seeking proofs.
Circle-in-circle halting study. click here
Circle-in-circle halting movies. click here
Square-in-square halting movies (periodic boundaries). click here
One dimension. click here
Rectangular ring chains in 3D. click here
Wallpaper symmetry. click here
Proximity trees. click here
A very sparse shape. click here
Measure zero. click here
An alternative area rule (and some odd mathematics). click here
The "holes" in a circle pattern have interesting features click here
How can one find the nearest neighbors of a given shape? click here This study led to the discovery of a rather odd form of highly-constrained random graph.
The statistical distribution of nearest-neighbor distances in a circle fractal has been studied. click here
The effect of shape on "packability" has been studied. click here
Papers with an art emphasis jointly authored by John and Prof. Doug Dunham (U. of Minn. Duluth) have been presented at the conferences Bridges 2014, JMM 2015, and Bridges 2015. Copies can be found at Doug Dunham's web site.