  Leja points on a compact $K \subset \mathbb{C}$ are known to provide
efficient points for interpolation, but their actual implementation can be
computationally challenging. So-called pseudo Leja points are a more tractable
solution, yet they require a tailored implementation to the compact at hand. We
introduce several more flexible random alternatives, starting from a new family
we call random Leja points. To make them tractable, we propose an approximate
version which relies on the Metropolis-Hastings algorithm with the uniform
measure. We also analyse a different family of points inspired by recently
introduced randomised admissible meshes, obtained by uniform sampling. When the
number of iterations or drawn points is appropriately chosen, we establish that
the two resulting families of points provide good points for interpolation.
That is, they almost surely lead to convergent interpolating polynomials for
holomorphic functions. The two last families of points are readily implemented
assuming one knows how to sample uniformly at random in $K$. These makes them
more modular than competing deterministic methods. We run numerical experiments
to compare the proposed methods in terms of accuracy and computational
complexity, for various types of compact sets.
