NEW CRITICAL BEHAVIOR IN d = 0 LARGE-N MATRIX MODELS

Abstract

The non-perturbative formulation of 2-dimensional quantum gravity in terms of the large-N limit of matrix models is studied to include the effects of higher order curvature terms. This leads to matrix models whose potential contains a symmetry breaking term of the form Tr ϕAϕA, where A is a given fixed matrix. This is studied in d = 0 dimensions and effectively induces additional terms of the form ( Tr ϕk)2 in the one matrix potential. An exact solution to leading order of the potential V(ϕ) = 1/2 Tr ϕ2 + g/N Tr ϕ4 + g′/N2 ( Tr ϕ2)2 is presented leading to 3 phases: γ = −1/2 (smooth surfaces), γ = 1/2 (branched polymer) and γ = 1/3 (intermediate phase). Including a Tr ϕ6 term in the potential gives rise to an additional phase with γ = 1/4. It is conjectured that for the general polynomial potential there are phases with γ = 1/n, n = 2, 3, …. The γ > 0 phases may correspond to c > 1 matter coupled to 2-dimensional gravity.