# 4.13 Statistical Mechanics

Features like dimension and curvature can be used to probe the consistent large-scale structure of the limiting behavior of our models. But particularly insofar as our models generate apparent randomness, it also makes sense to study their statistical features. We discussed above the overall distribution of values of Vr(X) and Δr(X). But we can also consider fluctuations and correlations.

For example, we can look at a 2-point correlation function Sr(s) = (Vr(X)Vr(Y) Vr(X)2)/Vr(X)2 for points X and Y separated by graph distance s. For a uniform graph such as the torus graph, Sr(s) always vanishes. For the buckyball approximation to the sphere that we used above, Sr(s) shows peaks at the distances between “pentagons” in the graph.

For the rule {{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}, Sr(s) steadily expands the region of s over which it shows positive correlations, and perhaps (at least for larger r, indicated by redder curves) approaches a limiting form:

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It is conceivable that for this or other rules there might be systematic rescalings of distance and number of steps that would lead to fixed limiting forms.

In statistical mechanics, it is common to think about the ensemble of all possible states of a systemand for example to discuss evolution from all possible initial conditions. But typical systems in statistical mechanics can basically be discussed in terms of a fixed number of degrees of freedom (either coordinates or values).

For our models, there is no obvious way to apply the rules but, for example, to limit the total number of relationsmaking it difficult to do analysis in terms of ensembles of states.

One can certainly imagine the set of all possible hypergraphs (and even have Ramsey-theory-style results about it), but this set does not appear to have the kind of geometry or structure that has typically been necessary for results in statistical mechanics or dynamical systems theory. (One could however potentially think in terms of a distribution of adjacency matrices, limiting to graphon-like functions  for infinite graphs.)