Random Vector and Matrix Theories: A Renormalization Group Approach

Abstract

The study of the statistical properties of random matrices of large size has a long history, dating back to Wigner, who suggested using Gaussian ensembles to give a statistical description of the spectrum of complex Hamiltonians and derived the famous semi-circle law, the work of Dyson and many others. Later, ’t Hooft noticed that, in \(SU(N)\) non-Abelian gauge theories, tessalated surfaces can be associated to Feynman diagrams and that the large \(N\) expansion corresponds to an expansion in successive topologies. Following this observation, some times later, it was realized that some ensembles of random matrices in the large N expansion and the so-called double scaling limit could be used as toy models for quantum gravity: 2D quantum gravity coupled to conformal matter. This has resulted in a tremendous expansion of random matrix theory, tackled with increasingly sophisticated mathematical methods and number of matrix models have been solved exactly. However, the somewhat paradoxical situation is that either models can be solved exactly or little can be said. Since the solved models display critical points and universal properties, it is tempting to use renormalization group ideas to determine universal properties, without solving models explicitly. Initiated by Brézin and Zinn-Justin, the approach has led to encouraging results, first for matrix integrals and then quantum mechanics with matrices, but has not yet become a universal tool as initially hoped. In particular, general quantum field theories with matrix fields require more detailed investigations. To better understand some of the encountered difficulties, we first apply analogous ideas to the simpler \(O(N)\) symmetric vector models, models that can be solved quite generally in the large \(N\) limit. Unlike other attempts, our method is a close extension of Brézin and Zinn-Justin. Discussing vector and matrix models with similar approximation scheme, we notice that in all cases (vector and matrix integrals, vector and matrix path integrals in the local approximation), at leading order, non-trivial fixed points satisfy the same universal algebraic equation, and this is the main result of this work. However, its precise meaning and role have still to be better understood.