  We study here a detailed conjecture regarding one of the most important cases
of anomalous diffusion, i.e the behavior of the "ant in the labyrinth". It is
natural to conjecture (see [16] and [8]) that the scaling limit for random
walks on large critical random graphs exists in high dimensions, and is
universal. This scaling limit is simply the natural Brownian Motion on the
Integrated Super-Brownian Excursion. We give here a set of four natural
sufficient conditions on the critical graphs and prove that this set of
assumptions ensures the validity of this conjecture. The remaining future task
is to prove that these sufficient conditions hold for the various classical
cases of critical random structures, like the usual Bernoulli bond percolation,
oriented percolation, spread-out percolation in high enough dimension. In the
companion paper [10], we do precisely that in a first case, the random walk on
the trace of a large critical branching random walk. We verify the validity of
these sufficient conditions and thus obtain the scaling limit mentioned above,
in dimensions larger than 14.
