  We develop numerical tools for Diagrammatic Monte-Carlo simulations of
non-Abelian lattice field theories in the t'Hooft large-N limit based on the
weak-coupling expansion. First we note that the path integral measure of such
theories contributes a bare mass term in the effective action which is
proportional to the bare coupling constant. This mass term renders the
perturbative expansion infrared-finite and allows to study it directly in the
large-N and infinite-volume limits using the Diagrammatic Monte-Carlo approach.
On the exactly solvable example of a large-N O(N) sigma model in D=2 dimensions
we show that this infrared-finite weak-coupling expansion contains, in addition
to powers of bare coupling, also powers of its logarithm, reminiscent of
re-summed perturbation theory in thermal field theory and resurgent
trans-series without exponential terms. We numerically demonstrate the
convergence of these double series to the manifestly non-perturbative dynamical
mass gap. We then develop a Diagrammatic Monte-Carlo algorithm for sampling
planar diagrams in the large-N matrix field theory, and apply it to study this
infrared-finite weak-coupling expansion for large-N U(N)xU(N) nonlinear sigma
model (principal chiral model) in D=2. We sample up to 12 leading orders of the
weak-coupling expansion, which is the practical limit set by the increasingly
strong sign problem at high orders. Comparing Diagrammatic Monte-Carlo with
conventional Monte-Carlo simulations extrapolated to infinite N, we find a good
agreement for the energy density as well as for the critical temperature of the
"deconfinement" transition. Finally, we comment on the applicability of our
approach to planar QCD at zero and finite density.
