We construct and study a matrix model that describes two dimensional string theory in the Euclidean black hole background. A conjecture of V. Fateev, A. and Al. Zamolodchikov, relating the black hole background to condensation of vortices (winding modes around Euclidean time) plays an important role in the construction. We use the matrix model to study quantum corrections to the thermodynamics of two dimensional black holes.

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The correspondence claimed by M. Douglas between the multicritical regimes of the two-matrix model and 2d gravity coupled with (p, q) rational matter field, is worked out explicitly. We found the minimal (p, q) multicritical potentials U (X) and V (Y ), which are polynomials of degree p and q, correspondingly. The loop averages W (X) andW (Y ) are shown to satisfy the Heisenberg relations {W, X} = 1 and {W , Y } = 1 and essentially coincide with the canonical momenta P and Q. The operators X and Y create the two kinds of boundaries in the (p, q) model related by the duality (p, q) ↔ (q, p). Finally, we present a closed expression for the two two-loop correlators, and interpret its scaling limit.

We study the regularized correlation functions of the light-like coordinate operators in the reduction to zero dimensions of the matrix model describing D-particles in four dimensions. We investigate in great detail the related matrix model originally proposed and solved in the planar limit by J. Hoppe. It also gives the solution of the problem of 3-coloring of planar graphs. We find interesting strong/weak 't Hooft coupling dependence.The partition function of the grand canonical ensemble turns out to be a tau-function of KP hierarchy. As an illustration of the method we present a new derivation of the large-N and double-scaling limits of the one-matrix model with cubic potential.