  How much time does it take for a fluctuating system, such as a polymer chain,
to reach a target configuration that is rarely visited -- typically because of
a high energy cost ? This question generally amounts to the determination of
the first-passage time statistics to a target zone in phase space with lower
occupation probability. Here, we present an analytical method to determine the
mean first-passage time of a generic non-Markovian random walker to a rarely
visited threshold, which goes beyond existing weak-noise theories. We apply our
method to polymer systems, to determine (i) the first time for a flexible
polymer to reach a large extension, and (ii) the first closure time of a stiff
inextensible wormlike chain. Our results are in excellent agreement with
numerical simulations and provide explicit asymptotic laws for the mean
first-passage times to rarely visited configurations.
