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

We study $d$-dimensional scalar field theory in the Local Potential Approximation of the functional renormalization group. Sturm-Liouville methods allow the eigenoperator equation to be cast as a Schrodinger-type equation. Combining solutions in the large field limit with the Wentzel-Kramers-Brillouin approximation, we solve analytically for the scaling dimension $d_n$ of high dimension potential-type operators ${\mathcal{O}}_n(\phi )$ around a non-trivial fixed point. We find that $d_n=n(d-d_{\phi })$ to leading order in $n$ as $n\to \mathrm{\infty }$, where $d_{\phi }=\frac{1}{2}(d-2+\eta )$ is the scaling dimension of the field, $\phi$, and determine the power-law growth of the subleading correction. For $O(N)$ invariant scalar field theory, the scaling dimension is just double this, for all fixed $N\ge 0$ and additionally for $N=-2,-4,\dots \mathrm{.}$ These results are universal, independent of the choice of cutoff function which we keep general throughout, subject only to some weak constraints.