[Networks generated by] random replacements

As indicated in the note above, applying the second rule (T1, shown as (b) on page 511) at an appropriate sequence of positions can transform one planar network into any other with the same number of nodes. The pictures below show what happens if this rule is repeatedly applied at random positions in a network. Each time it is applied, the rule adds two edges to one face, and removes them from another. After many steps the pictures below show that faces with large numbers of edges appear. The average number of edges must always be 6 (see note above), but in a sufficiently large network the probability for a face to have n edges eventually approaches an equilibrium value of 8 (n - 2)(2n - 3)!! (3/8)ⁿ/n!. (For large n this is approximately λⁿ with λ = 3/4; if 1- and 2-edged regions are allowed then λ = (3 + √3)/6 ≃ 0.79.) There may be some easy way to derive such results, but so far it has only been done using fairly sophisticated techniques from quantum field theory developed in the late 1970s. The starting point is to look at a φ³ field theory with SU(n) internal symmetry and to note that in the limit n  ∞ what dominates are Feynman diagrams that have the structure of planar trivalent networks (see page 1040). And it then turns out that in zero spacetime dimensions the complete path integral for the theory can be evaluated exactly—yielding in effect a generating function for the number of possible networks. Parametric differentiation (to yield n-point correlation functions) then gives results for n-sided regions. Another result that has been derived is that the average total number m[n] of edges of all faces around a given face with n edges is 7n + 3 + 9/(n + 1). Note that the networks obtained always have dimension 2 according to my definitions.