Doubling of equations and universality in matrix models of random surfaces

Abstract

We prove that in the recently proposed scaling limit [E. Brézin and V.A. Kazakov, Phys. Lett. B 236 (1990) 144; M.R. Douglas and S. Shenker, Rutgers preprint RU-89/34 (October 1989); D.J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127] in matrix models of random surfaces, the (singular piece of the) free energy is obtained from the sun of solutions of two non-linear differential equations: ϵ ±z= D (m) (f±g) . These are identical and universal modulo the two arbitrary parameters ϵ ±, and the (common) normalization of the string coupling z. The doubling of equations implies a doubling of non-perturbative parameters. For even matrix-potentials one of the non-universal constants is fixed: ϵ += ϵ −, and the scaling function g vanishes to all orders in the loop expansion.