Archive for December, 2009

Dimidium facti qui coepit habet

December 12, 2009

Palmyra  (by loufi on flickr).

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Brownian motion and nonabelian finite groups

December 11, 2009

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

Let G be a non-abelian finite group, and G' its derived group.  Let Pr(G) be the probability that two elements of G commute: Pr(G):=N_c(G)/N_t(G),  where N_c(G) is the number of ordered pairs (x,y)\in G\times G such that xy=yx,   and N_t(G)  is the total number of ordered pairs in G\times G.

Rusin has shown that Pr(G) \leq \frac{1}{4}+\frac{3}{4}\frac{1}{|G'|}.  Then the maximum value of Pr(G) is 5/8, and is attained for G=H_8 and G=D_4, two groups of order 8. Moreover, when |G'|<8 we have Pr(G)>11/32.

Now to another world where these very numbers also appear.   For any planar lattice L, denote by a_{n,L} the number of self-avoiding nearest-neighbour paths, and call \mu_L:=\inf_{n\geq 1}(a_{n,L})^{1/n} the connectivity constant.

Then a famous conjecture of Nienhuis is that for any lattice L one has a_n(L)=(\mu_L)^nn^{11/32+o(1)} as n\rightarrow +\infty. This rate has been proven conditionally by Lawler-Schramm-Werner.

Also, these authors had previously shown the following result: let B and B' be two independent Brownian motions with |B_0-B_0'|=1, then there exists a constant c such that  the probability that these two random paths do not intersect is governed by the exponent 5/8, namely one has t^{-5/8}/c\leq P[ B(0,t) \cap B'(0,t)=\emptyset ]\leq c t^{-5/8}. (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?

Commuting elements of non-abelian groups

December 11, 2009

The following is simply a collection of results, which I haven’t checked  just yet.

Let G be a finite non-abelian group.  Let N_c(G) be the number of ordered pairs (x,y)\in G\times G such that xy=yx,   and N_t(G)  be the total number of ordered pairs in G\times G.  So the real number Pr(G):=N_c(G)/N_t(G) belongs to [0,1].

In 1979, Rusin has worked out explicitely the groups for which Pr(G)>11/32, in this freely available paper.  The maximum value is 5/8, attained both by the quaternion group H_8 and the dihedral group D_4.   Then there is an infinite decreasing series starting at 5/8 and accumulating at 1/2.  And finally between 1/2 and 11/32 there are a handful of other cases.

Some problems regarding lower bounds, or the density of the values of Pr(G) in intervals [a,b] with b\leq 1/4 are mentionned, I believe they are still open.

The problem has been generalized more recently:   to n-tuples in compact infinite groups by Erfanian and Russo,   to pairs of  subgroups by Tărnăuceanu,  and to semigroups by Givens (in which case there is density in [0,1].)