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

Feasibility of Diagrammatic Monte-Carlo based on weak-coupling expansion  in asymptotically free theories: case study of $O(N)$ sigma-model in the  large-$N$ limit