Two-dimensional Lorentzian Models

The goal of this paper is to present rigorous mathematical formulations and results for Lorentzian models, introduced in physical papers. Lorentzian models represent two dimensional models, where instead of a two-dimensional lattice one considers an ensemble of triangulations of a cylinder, and natural probability measure (Gibbs family) on this ensemble. It appears that correlation functions of this model can be found explicitly. Such models can be considered as an example of a new approach to quantum gravity, based on the notion of a causal set. Causal set is a partially ordered set, thus having a causal structure, similar to Minkowski space. We consider subcritical, critical and supercritical cases. In the critical case the scaling limit of the light cone can be restored. 2000 Math. Subj. Class. 83C27, 82B41, 37K99.