  Diffusion processes with small noise conditioned to reach a target set are
considered. The AMS algorithm is a Monte Carlo method that is used to sample
such rare events by iteratively simulating clones of the process and selecting
trajectories that have reached the highest value of a so-called importance
function. In this paper, the large sample size relative variance of the AMS
small probability estimator is considered. The main result is a large
deviations logarithmic equivalent of the latter in the small noise asymptotics,
which is rigorously derived. It is given as a maximisation problem explicit in
terms of the quasi-potential cost function associated with the underlying small
noise large deviations. Necessary and sufficient geometric conditions ensuring
the vanishing of the obtained quantity ('weak' asymptotic efficiency) are
provided. Interpretations and practical consequences are discussed.
