  We propose a method for determining the exact correspondence between the
Wilsonian cut-off scale on the boundary and its holographically dual bulk
theory. We systematically construct the multi-trace Wilsonian effective action
from holographic renormalisation and evolve it by integrating out the
asymptotically Anti-de Sitter bulk geometry with scalar probes. The Wilsonian
nature of the effective action is shown by proving that it must be either
double-trace, closing in on itself under successive integrations, or have an
infinite series of multi-trace terms. Focusing on composite scalar operator
renormalisation, we relate the Callan-Symanzik equation, the flow of the scalar
anomalous dimension and the multi-trace beta functions to their dual RG flows
in the bulk. Establishing physical renormalisation conditions on the behaviour
of the large-$N$ anomalous dimension then enables us to extract the energy
scales. Examples of pure AdS, GPPZ flow, black brane in AdS, M2 and M5 branes
are studied before we generalise our results to arbitrary numbers of mass and
thermal deformations of an ultra-violet CFT. Relations between the undeformed
Wilsonian cut-off, deformation scales and the deformed Wilsonian cut-off are
discussed, as is phenomenology of each considered background. We see how a mass
gap, the emergent infra-red CFT scaling, etc. arise in different effective
theories. We also argue that these results can have alternative interpretations
through the flow of the conformal anomaly or the Ricci scalar curvature of
boundary branes. They show consistency with the c-theorem.
