  The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile
method for the simulation of rare events. It is based on an interacting (via a
mutation-selection procedure) system of replicas, and depends on two integer
parameters: n $\in$ N * the size of the system and the number k $\in$ {1, . . .
, n -- 1} of the replicas that are eliminated and resampled at each iteration.
In an idealized setting, we analyze the performance of this algorithm in terms
of a Large Deviations Principle when n goes to infinity, for the estimation of
the (small) probability P(X \textgreater{} a) where a is a given threshold and
X is real-valued random variable. The proof uses the technique introduced in
[BLR15]: in order to study the log-Laplace transform, we rely on an auxiliary
functional equation. Such Large Deviations Principle results are potentially
useful to study the algorithm beyond the idealized setting, in particular to
compute rare transitions probabilities for complex high-dimensional stochastic
processes.
