The probability that $x^m$ and $y^n$ commute in a compact group

In a recent article [K.H. Hofmann and F.G. Russo, The probability that $x$ and $y$ commute in a compact group, Math. Proc. Cambridge Phil. Soc., to appear] we calculated for a compact group $G$ the probability $d(G)$ that two randomly picked elements $x, y\in G$ satisfy $xy=yx$, and we discussed the remarkable consequences on the structure of $G$ which follow from the assumption that $d(G)$ is positive. In this note we consider two natural numbers $m$ and $n$ and the probabilty $d_{m,n}(G)$ that for two randomly selected elements $x, y\in G$ the relation $x^my^n=y^nx^m$ holds. The situation is more complicated whenever $n,m>1$. If $G$ is a compact Lie group and if its identity component $G_…