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 φ^k)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.