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 $n\to\infty$ for a constant $A$, which we will give explicitly. 
Misja F.A. Steinmetz, Madeleine L. Whybrow
math.GR