Asypmtotic Behaviour of the Conjugacy Probability of the Alternating Group

For $G$ a finite group, let $\kappa (G)$ be the probability that $\sigma ,\tau \in G$ are conjugate, when $\sigma$ and $\tau$ are chosen independently and uniformly at random. In this paper, we extend the methods of Blackburn et al [2012] to show that $n^2\kappa (A_n)\to A$ as $n\to \mathrm{\infty }$ for a constant $A$, which we will give explicitly.