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.

We consider the system of particles with equal charges and nearest neighbour Coulomb interaction on the interval. We study local properties of this system, in particular the distribution of distances between neighbouring charges. For zero temperature case there is sufficiently complete picture and we give a short review. For Gibbs distribution the situation is more difficult and we present two related results.

. Each particle moves with constant speed, initially prescribed to it. When particle and antiparticle collide, they both disappear. It is the only interaction in the system. We find explicitly the large time asymptotics of β t -the coordinate of the last collision before t between particle and antiparticle.

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