Archive for December, 2009

Dimidium facti qui coepit habet

December 12, 2009

Palmyra  (by loufi on flickr).

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].)