  We discuss the feasibility of applying Diagrammatic Monte-Carlo algorithms to
the weak-coupling expansions of asymptotically free quantum field theories,
taking the large-$N$ limit of the $O(N)$ sigma-model as the simplest example
where exact results are available. We use stereographic mapping from the sphere
to the real plane to set up the perturbation theory, which results in a small
bare mass term proportional to the coupling $\lambda$. Counting the powers of
coupling associated with higher-order interaction vertices, we arrive at the
double-series representation for the dynamically generated mass gap in powers
of both $\lambda$ and $\log(\lambda)$, which converges quite quickly to the
exact non-perturbative answer. We also demonstrate that it is feasible to
obtain the coefficients of these double series by a Monte-Carlo sampling in the
space of Feynman diagrams. In particular, the sign problem of such sampling
becomes milder at small $\lambda$, that is, close to the continuum limit.
