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.
Highlights
Random matrix models are specific statistical models describing (Euclidean) quantum fluctuations of a matrixlike field [1]
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
The link between matrix models and two-dimensional quantum gravity arises from the observation that the perturbation series of random matrix models can generate randomly arbitrary triangulated surfaces; the precise relation between Feynman diagrams and elementary polygons being discussed on a concrete example in Sec
Summary
Random matrix models are specific statistical models describing (Euclidean) quantum fluctuations of a matrixlike field [1]. A nonperturbative FRG framework has been considered to improve the perturbative results [12] In this reference paper, the authors show convergence phenomena for the computed critical exponents toward the exact (i.e., analytic result) for double scaling. Due to the presence of the regulator, the compatibility with Ward identities requires to enlarge the theory space to derivative couplings; which in turn seems to play a significant role in the fixed point structure, and introduce a spurious dependence on the regulator To solve this issue, we introduce a modified regulator, parametrized in such a way that the contribution of derivative couplings in the Ward identities remains small in a significant domain of the RG flow, so that truncation involving only traces may be used without strong disagreements to approximate the exact solution of the RG equations. V we provide some discussions and the conclusion of this work
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