Relational background field formulation of discrete quantum gravity models: The case of $O(N)$ vector models in zero dimensions

Abstract

In a recent paper arxiv.org/abs/2609.21031, we have introduced the concept of a relational background field as a new approach to the renormalization group (RG) for discrete quantum gravity models, focusing on random matrix models. The proposed method offers the dual advantage of preserving the model's gauge symmetry (typically $U(N)$ or $O(N)$), whose breaking in conventional approaches has remained an open problem, and providing a more satisfactory treatment of the continuum or infrared (IR) limit. In this limit, the theory is described by a non-local field theory in dimension $D = 2k + 3$, where $k$ denotes the number of cuts in the effective spectrum as dictated by random matrix theory. This pedagogical paper reviews this construction for real random vector models in zero dimensions, with $O(N)$ as the symmetry group. Following the general strategy of background-field-type formulations, we show, using the BBP (Baik-Ben Arrous-P'ech'e) phase transition, that a partial Hubbard--Stratonovich decomposition of the quartic interaction, involving a matrix-valued intermediate field, allows a preferred notion of scale to emerge. This scale, in turn, defines an RG flow, whose perturbative structure is analyzed. Our calculation demonstrates the existence of an \textit{asymptotic} Wilson--Fisher-type fixed point, whose sole relevant critical exponent is in qualitative agreement with the double-scaling critical exponent of these models.