Growth of uniform infinite causal triangulations

TL;DR

This paper introduces a growth process for uniform infinite causal triangulations, relates it to a random walk, and analyzes geometric properties like fractal dimension and boundary behavior.

Contribution

It presents a novel growth model for causal triangulations and establishes connections to random walks, diffusion processes, and branching processes.

Findings

Relation between growth process and random walk on half line

Estimation of geodesic distance and fractal dimension

Convergence of boundary process to a diffusion process

Abstract

We introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added. A relation to a random walk on the integer half line is shown. This relation is used to estimate the geodesic distance of a given triangle to the rooted boundary in terms of the time of the growth process and to determine from this the fractal dimension. Furthermore, convergence of the boundary process to a diffusion process is shown leading to an interesting duality relation between the growth process and a corresponding branching process.