  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.
