  A construction as a growth process for sampling of the uniform infinite
planar triangulation (UIPT), defined in a previous paper, is given. The
construction is algorithmic in nature, and is an efficient method of sampling a
portion of the UIPT.
  By analyzing the progress rate of the growth process we show that a.s. the
UIPT has growth rate r^4 up to polylogarithmic factors, confirming heuristic
results from the physics literature. Additionally, the boundary component of
the ball of radius r separating it from infinity a.s. has growth rate r^2 up to
polylogarithmic factors. It is also shown that the properly scaled size of a
variant of the free triangulation of an m-gon converges in distribution to an
asymmetric stable random variable of type 1/2.
  By combining Bernoulli site percolation with the growth process for the UIPT,
it is shown that a.s. the critical probability p_c=1/2 and that at p_c
percolation does not occur.
