A Sequential Growth Dynamics for a Directed Acyclic Dyadic Graph

A model of discrete spacetime on a microscopic level is considered. It is a directed acyclic dyadic graph (an x-graph). The dyadic graph means that each vertex possesses no more than two incident incoming edges and two incident outgoing edges. This model is the particular case of a causal set because the set of vertices of x-graph is a causal set. The sequential growth dynamics is considered. This dynamics is a stochastic sequential additions of new vertices one by one. A new vertex can be connected with existed vertex by an edge only if the existed vertex possesses less than four incident edges. There are four types of such additions. The probabilities of different variants of addition of a new vertex depend on the structure of existed x-graph. These probabilities are the functions of the probabilities of random choice of directed paths in the x-graph. The random choice of directed paths is based on the binary alternatives. In each vertex of the directed path we choose one of two possible edges to continue this path. It is proved that such algorithm of the growth is a consequence of a causality principle and some conditions of symmetry and normalization. The probabilities are represented in a matrix form. The iterative procedure to calculate probabilities is considered. Elementary evolution operators is introduced. The second variant to calculate probabilities is based on these elementary evolution operators.

causal set, random graph, directed graph