  Perturbation theory in asymptotically free quantum field theories is
asymptotic. The factorially growing perturbative coefficients carry information
about non-perturbative corrections, which can be related to renormalons and
instantons. Using the Wiener-Hopf technique we determine the full analytic
solution for the free energy density in the two dimensional $O(N)$ sigma
models. For $N>3$ there are no instantons, and we found that the perturbative
series carries all the information about the non-perturbative corrections.
However, in the $O(3)$ case, we identify several non-perturbative sectors that
are not related to the asymptotics of the perturbative series. The number of
sectors depends on the observables: for the ground-state energy density we
identify three sectors, which we attribute to instantons. For the free energy
density in the running perturbative coupling we found infinitely many sectors.
