Numerical bounds for the exterior degree of finite simple groups

he exterior degree $d^\wedge(G)$ of a finite group $G$ is the probability that a pair $(x,y)$ of elements $x,y$ chosen uniformly at random in $G$ satisfies the condition $x \wedge y=1_\wedge$, where $\wedge$ denotes the operator of nonabelian exterior square $G \wedge G$ and $1_\wedge$ is the neutral element in $G\wedge G$. The well known probability $d(G)$ that two elements of $G$ commute (in the sense of Erd\H{o}s and Tur\'an) upper bounds $d^\wedge(G)$. Of course, $d(G)=1$ if and only if $G$ is abelian, but $d^\wedge(G) =1$ if and only if $G$ is cyclic, so we may detect cyclic groups as long as $d^\wedge(G)$ is close to $1$ (among abelian groups). We present new numerical results for the exterior degree of finite simple groups.