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The generating functional $\mathcal{W}[J_{\mathcal O}]$ of Euclideancorrelators of twist-$2$ operators in SU($N$) Yang-Mills theory admits the 'tHooft large-$N$ expansion: $\mathcal{W}[J_{\mathcalO}]=\mathcal{W}_{sphere}\,\,\,\,[J_{\mathcal O}]+\mathcal{W}_{torus}\,\,\,[J_{\mathcal O}]+ \cdots$. Nonperturbatively, $\mathcal{W}_{sphere}\,\,\,\,[J_{\mathcal O}]$ is a sum of tree diagrams involving glueballpropagators and vertices, while $\mathcal{W}_{torus} \,\,\,[J_{\mathcal O}]$ isa sum of glueball one-loop diagrams. Moreover, it has been predicted that$\mathcal{W}_{torus } \,\,\,[J_{\mathcal O}]$ should admit the structure of thelogarithm of a functional determinant summing glueball one-loop diagrams. Wework 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 allthe coordinates of the correlators are uniformly rescaled by a factor $\lambda\rightarrow 0$. Remarkably, we verify the above prediction that$\mathcal{W}_{asym \, torus} \,\,\,\,\,\,[J_{\mathcal O},\lambda]$ -- beingasymptotic 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 thenonperturbative solution of large-$N$ YM theory and it may be a pivotal guidefor the search of such a solution.

UV asymptotics of $n$-point correlators of twist-$2$ operators in SU($N$) Yang-Mills theory