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We study $d$-dimensional scalar field theory in the Local PotentialApproximation of the functional renormalization group. Sturm-Liouville methodsallow the eigenoperator equation to be cast as a Schrodinger-type equation.Combining solutions in the large field limit with the Wentzel-Kramers-Brillouinapproximation, we solve analytically for the scaling dimension $d_n$ of highdimension potential-type operators $\mathcal{O}_n(\varphi)$ around anon-trivial fixed point. We find that $d_n = n(d-d_\varphi)$ to leading orderin $n$ as $n \to \infty$, where $d_\varphi=\frac{1}{2}(d-2+\eta)$ is thescaling dimension of the field, $\varphi$, and determine the power-law growthof the subleading correction. For $O(N)$ invariant scalar field theory, thescaling dimension is just double this, for all fixed $N\geq0$ and additionallyfor $N=-2,-4,\ldots \,.$ These results are universal, independent of the choiceof cutoff function which we keep general throughout, subject only to some weakconstraints.

Universal scaling dimensions for highly irrelevant operators in the Local Potential Approximation