Phases of the large- N matrix model and non-perturbative effects in 2D gravity

Abstract

The large- N solution of the one-matrix model of E. Brézin, C. Itzykson, G. Parisi and J.-B. Zuber is reconsidered for generic complex potential. A regular large- N limit does not exist in some singular domain, which depends on the prescription chosen in order to make the matrix integral convergent at infinity. Near the m = 2 critical point the singular domain (in the scaling variable x complex plane) is a sector of angle 2π 5 coinciding with the sector of poles of the “triply truncated” Painlevé I transcendent of Boutroux, which is therefore (although not real) the only solution of the string equation compatible with the matrix model and the loop equations for two-dimensional gravity. Our approach allows us to relate non-perturbative effects in the string equations to instantons in the matrix model and to discuss the flows between multicritical points.