  We initiate a systematic, non-perturbative study of the large-$N$ expansion
in the two-dimensional $\text{SU}(N)\times \text{SU}(N)$ Principal Chiral Model
(PCM). Starting with the known infinite-$N$ solution for the ground state at
fixed chemical potential \cite{Fateev:1994dp,Fateev:1994ai}, we devise an
iterative procedure to solve the Bethe ansatz equations order by order in
$1/N$. The first few orders, which we explicitly compute, reveal a systematic
enhancement pattern at strong coupling calling for the near-threshold
resummation of the large-$N$ expansion. The resulting double-scaling limit
bears striking similarities to the $c=1$ non-critical string theory and
suggests that the double-scaled PCM is dual to a non-critical string with a
$(2+1)$-dimensional target space where an additional dimension emerges
dynamically from the $\text{SU}(N)$ Dynkin diagram.
