Space as a ''bucket of dust'' (Q1058174)

Wheeler has suggested the need for pre-geometry, that is, a prior structure from which continuous space can be constructed (at least approximately). This paper takes seriously the idea of starting with a large number of elements which can be connected; it claims that with very few assumptions such a set will exhibit properties usually associated with a three-dimensional space of constant positive curvature. It assumes that the elements are the vertices of a random graph, with each edge having constant small probability p of existing; distance is then defined in the usual graph-theoretic way. The probability distribution of the ''shape'' of the universe (that is, the assemblage of distances) is then found, and a rather complex analysis of the most probable distribution shows a ln N/ln L relation \((N=number\) of elements, \(L=size)\) approaching 3, independent of p so long as it is small. The result appears to depend critically on p being both small and constant.