  We study the large source asymptotics of the generating functional in quantum
field theory using the holographic renormalization group, and draw comparisons
with the asymptotics of the Hopf characteristic function in fractal geometry.
Based on the asymptotic behavior, we find a correspondence relating the Weyl
anomaly and the fractal dimension of the Euclidean path integral measure. We
are led to propose an equivalence between the logarithmic ultraviolet
divergence of the Shannon entropy of this measure and the integrated Weyl
anomaly, reminiscent of a known relation between logarithmic divergences of
entanglement entropy and a central charge. It follows that the information
dimension associated with the Euclidean path integral measure satisfies a
c-theorem.
