  Using a novel theoretical approach, we study the mean first encounter time
(MFET) between the two ends of a polymer. Previous approaches used various
simplifications that reduced the complexity of the problem, leading, however to
incompatible results. We construct here for the first time a general theory
that allows us to compute the MFET. The method is based on estimating the mean
time for a Brownian particle to reach a narrow domain in the polymer
configuration space. In dimension two and three, we find that the MFET depends
mainly on the first eigenvalue of the associated Fokker-Planck operator and
provide precise estimates that are confirmed by Brownian simulations.
Interestingly, although many time scales are involved in the encounter process,
its distribution can be well approximated by a single exponential, which has
several consequences for modeling chromosome dynamics in the nucleus. Another
application of our result is computing the mean time for a DNA molecule to form
a closed loop (when its two ends meet for the first time).
