Renormalization group flow in one- and two-matrix models

TL;DR

This paper derives and analyzes renormalization group equations for one- and two-matrix models, revealing fixed points, scaling exponents, and phase structure relevant to two-dimensional quantum gravity.

Contribution

It presents a formulation of RG equations with a finite number of couplings and analyzes their fixed points and phase structure, connecting to quantum gravity.

Findings

Fixed points and scaling exponents match exact solutions.

Linearized beta functions approximate phase structure well.

Global flow suggests a c-theorem in 2D quantum gravity.

Abstract

Large-$N$ renormalization group equations for one- and two-matrix models are derived. The exact renormalization group equation involving infinitely many induced interactions can be rewritten in a form that has a finite number of coupling constants by taking account of reparametrization identities. Despite the nonlinearity of the equation, the location of fixed points and the scaling exponents can be extracted from the equation. They agree with the spectrum of relevant operators in the exact solution. A linearized $\beta$-function approximates well the global phase structure which includes several nontrivial fixed points. The global renormalization group flow suggests a kind of $c$-theorem in two-dimensional quantum gravity.