The inefficiency of re-weighted sampling and the curse of system size in
high order path integration

Computing averages over a target probability density by statistical
re-weighting of a set of samples with a different distribution is a strategy
which is commonly adopted in fields as diverse as atomistic simulation and
finance. Here we present a very general analysis of the accuracy and efficiency
of this approach, highlighting some of its weaknesses. We then give an example
of how our results can be used, specifically to assess the feasibility of
high-order path integral methods. We demonstrate that the most promising of
these techniques -- which is based on re-weighted sampling -- is bound to fail
as the size of the system is increased, because of the exponential growth of
the statistical uncertainty in the re-weighted average.