Revisited functional renormalization group approach for random matrices in the large-$N$ limit

TL;DR

This paper revisits the functional renormalization group approach for large-$N$ matrix models, analyzing approximation methods, Ward identity compatibility, and regulator effects to improve understanding of fixed points and critical exponents.

Contribution

It critically examines the local potential approximation and introduces derivative couplings to better align with Ward identities in matrix models.

Findings

Standard local potential approximation violates Ward identities near fixed points.

Including derivative couplings recovers an interacting fixed point with improved critical exponents.

Modified regulator improves flow behavior and results alignment with existing literature.

Abstract

The nonperturbative renormalization group has been considered as a solid framework to investigate fixed point and critical exponents for matrix and tensor models, expected to correspond with the so-called double scaling limit. In this paper, we focus on matrix models and address the question of the compatibility between the approximations used to solve the exact renormalization group equation and the modified Ward identities coming from the regulator. We show in particular that standard local potential approximation strongly violates the Ward identities, especially in the vicinity of the interacting fixed point. Extending the theory space including derivative couplings, we recover an interacting fixed point with a critical exponent not so far from the exact result, but with a nonzero value for derivative couplings, evoking a strong dependence concerning the regulator. Finally, we consider a modified regulator, allowing to keep the flow not so far from the ultralocal region and recover the results of the literature up to a slight improvement.