  The generating functional $\mathcal{W}[J_{\mathcal O}]$ of Euclidean
correlators of twist-$2$ operators in SU($N$) Yang-Mills theory admits the 't
Hooft large-$N$ expansion: $\mathcal{W}[J_{\mathcal
O}]=\mathcal{W}_{sphere}\,\,\,\,[J_{\mathcal O}]+\mathcal{W}_{torus}
\,\,\,[J_{\mathcal O}]+ \cdots$. Nonperturbatively, $\mathcal{W}_{sphere}
\,\,\,\,[J_{\mathcal O}]$ is a sum of tree diagrams involving glueball
propagators and vertices, while $\mathcal{W}_{torus} \,\,\,[J_{\mathcal O}]$ is
a sum of glueball one-loop diagrams. Moreover, it has been predicted that
$\mathcal{W}_{torus } \,\,\,[J_{\mathcal O}]$ should admit the structure of the
logarithm of a functional determinant summing glueball one-loop diagrams. We
work out in a closed form the ultraviolet (UV) asymptotics of
$\mathcal{W}_{sphere} \,\,\,\,[J_{\mathcal O},\lambda] \sim \mathcal{W}_{asym
\, sphere} \,\,\,\,\,\,\,[J_{\mathcal O},\lambda]$ and $\mathcal{W}_{torus}
\,\,\,[J_{\mathcal O},\lambda] \sim \mathcal{W}_{asym \, torus}
\,\,\,\,\,\,[J_{\mathcal O},\lambda]$ in the coordinate representation as all
the coordinates of the correlators are uniformly rescaled by a factor $\lambda
\rightarrow 0$. Remarkably, we verify the above prediction that
$\mathcal{W}_{asym \, torus} \,\,\,\,\,\,[J_{\mathcal O},\lambda]$ -- being
asymptotic in the UV to $\mathcal{W}_{torus} \,\,\,[J_{\mathcal O}, \lambda]$
-- admits the structure of the logarithm of a functional determinant as well.
Hence, the computation above sets strong UV asymptotic constraints on the
nonperturbative solution of large-$N$ YM theory and it may be a pivotal guide
for the search of such a solution.
