  We study oriented percolation on random causal triangulations, those are
random planar graphs obtained roughly speaking by adding horizontal connections
between vertices of an infinite tree. When the underlying tree is a geometric
Galton--Watson tree with mean $m>1$, we prove that the oriented percolation
undergoes a phase transition at $p_c(m)$, where $p_c(m) = \frac{\eta}{1+\eta}$
with $\eta = \frac{1}{m+1} \sum_{n \geq 0} \frac{m-1}{m^{n+1}-1}$. We establish
that strictly above the threshold $p_c(m)$, infinitely many infinite components
coexist in the map. This is a typical percolation result for graphs with a
hyperbolic flavour. We also demonstrate that large critical oriented
percolation clusters converge after rescaling towards the Brownian continuum
random tree. The proof is based on a Markovian exploration method, similar in
spirit to the peeling process of random planar maps.
