Based on the previous post, I can’t help make the following observations.

Let be a non-abelian finite group, and its derived group. Let be the probability that two elements of commute: , where is the number of ordered pairs such that , and is the total number of ordered pairs in .

Rusin has shown that . Then the maximum value of is , and is attained for and , two groups of order 8. Moreover, when we have .

Now to another world where these very numbers also appear. For any planar lattice , denote by the number of self-avoiding nearest-neighbour paths, and call the connectivity constant.

Then a famous conjecture of Nienhuis is that for any lattice one has as . This rate has been proven conditionally by Lawler-Schramm-Werner.

Also, these authors had previously shown the following result: let and be two independent Brownian motions with , then there exists a constant such that the probability that these two random paths do not intersect is governed by the exponent , namely one has . (Their results are vastly more general, but boil down to this for two families made by one Brownian each).

So, in both cases of non-abelian finite groups and of self-avoiding random walks we have these two rather noticeable numbers 5/8 and 11/32 appearing: is there something deep going on, or is this just an accident?

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