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Do we have (proposed?) methods to look for fixed points in the renormalization group flow of the Einstein-Hilbert action? My understanding of the RG is still somewhat sketchy at this point and I am having trouble understanding how one would go about searching for a fixed point in a theory that's non-renormalizable.

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  • $\begingroup$ I don't have good understanding of this yet but I think when people talk about RG flow in a gravity context, they mean holographic RG flow, which uses AdS/CFT duality to relate the classical solutions in AdS to RG flow of the boundary field theory. Thus fixed points should be understood in this context, instead of fixed points for a non-renormalizable theory. $\endgroup$
    – user110373
    Feb 29 '16 at 1:41
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Here are two papers on the subject:

https://arxiv.org/abs/0805.2909 Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation

From the abstract: We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation. We begin by discussing various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so-called Einstein-Hilbert truncation, where only the cosmological constant and Newton's constant are retained. In this context we make connection with old results, in particular we reproduce the results of the epsilon expansion and the perturbative one loop divergences.

https://arxiv.org/abs/1601.01800 The Gravitational Two-Loop Counterterm is Asymptotically Safe

Abstract: Weinberg's asymptotic safety scenario provides an elegant mechanism to construct a quantum theory of gravity within the framework of quantum field theory based on a non-Gau{\ss}ian fixed point of the renormalization group flow. In this work we report novel evidence for the validity of this scenario, using functional renormalization group techniques to determine the renormalization group flow of the Einstein-Hilbert action supplemented by the two-loop counterterm found by Goroff and Sagnotti. The resulting system of beta functions comprises three scale-dependent coupling constants and exhibits a non-Gau{\ss}ian fixed point which constitutes the natural extension of the one found at the level of the Einstein-Hilbert action. The fixed point exhibits two ultraviolet attractive and one repulsive direction supporting a low-dimensional UV-critical hypersurface. Our result vanquishes the longstanding criticism that asymptotic safety will not survive once a "proper perturbative counterterm" is included in the projection space.

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You can use functional renormalization group method to renormalize Quantum Einstein Gravity.You can see how to renormalize it in arXiv:hep-th/9605030.
If you have problems with understanding this paper, see the paragraph 'Exact renormalization groups' in article Renormalization group in Wikipedia. I wanted to post more links but I don't have enough reputation to post multiple links. Please excuse me.

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