  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 $\mathcal{O}_n(\varphi)$ around a
non-trivial fixed point. We find that $d_n = n(d-d_\varphi)$ to leading order
in $n$ as $n \to \infty$, where $d_\varphi=\frac{1}{2}(d-2+\eta)$ is the
scaling dimension of the field, $\varphi$, 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 $N\geq0$ and additionally
for $N=-2,-4,\ldots \,.$ These results are universal, independent of the choice
of cutoff function which we keep general throughout, subject only to some weak
constraints.
