The construction of a reliable renormalization group (RG) flow for discrete gravity models that preserves their underlying symmetry group, typically $U(N)$ or $O(N)$, remains an open problem. In this pedagogical presentation, we will show, using the example of a zero-dimensional$O(N)$ vector model, how a covariant renormalization group can be constructed using a background field approach. Here, this background field takes the form of a real Wigner matrix, obtained by decomposing a portion of the model's quartic interaction into a three-body vector-matrix interaction via a Hubbard-Stratonovich transformation. The concentration of the measure at large $N$, and the BBP (Baik-Ben Arous-Péché) theorem for the ordinary 1-spike matrix model, show that the background generically freezes onto a Wigner matrix and its spectrum. This provides the basis for a relational RG in which the notion of scale is dynamically induced by the effective Gaussian measure, in the basis where the intermediate field is diagonal. We show how flow equations can be effectively derived, and that there exists an asymptotic IR fixed point whose critical exponents agree with the analytical values deduced from the double scaling limit. This method has also been applied to random matrix models for 2D quantum gravity, and an extension to random tensors and group field theories will be discussed.