It is known that the probability in question exceeds a positive constant for $N>N_0$, and it is conjectured that it tends to $1-e^{-2}\approx 0.864665$ as $N\to\infty$. I suggest to read the introduction of this [paper by Teräväinen][1] for an overview of this research area.


  [1]: https://arxiv.org/abs/1510.06005