  We classify completely the infinite, planar triangulations satisfying a weak
spatial Markov property, without assuming one-endedness nor finiteness of
vertex degrees. In particular, the Uniform Infinite Planar Triangulation (UIPT)
is the only such triangulation with average degree 6. As a consequence, we
prove that the convergence of uniform triangulations of the sphere to the UIPT
is robust, in the sense that it is preserved under various perturbations of the
uniform measure. As another application, we obtain large deviation estimates
for the number of occurencies of a pattern in uniform triangulations.
