Path Integral Measure, Running Scale and RG flow for Gravity

We study the Wilsonian renormalization group (RG) flow of Euclidean quantum gravity in the Einstein-Hilbert truncation, paying attention to two important ingredients: the local path integral measure and the physical running scale. We introduce a numerical running scale L for the integration of the metric and ghost modes shell by shell, and consider two possible relations between L and the running scale k. In the first case, the numerical cut L is converted into the running scale k through the running de Sitter radius selected by the action at that scale. In the second one, it is converted through the radius of the off-shell spherical background. The two prescriptions lead to different beta functions and therefore to different fixed-point structures. This shows that, already within the same truncation and the same spectral organization of modes, the RG flow is sensitive to the geometrical meaning assigned to the RG scale k.