  We construct a measure valued Markov process which we call infinite canonical
super-Brownian motion, and which corresponds to the canonical measure of
super-Brownian motion conditioned on non-extinction. Infinite canonical
super-Brownian motion is a natural candidate for the scaling limit of various
random branching objects on $\Z^d$ when these objects are (a) critical; (b)
mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the
spread-out oriented percolation incipient infinite cluster above 4 dimensions
and of incipient infinite branching random walk in any dimension. We conjecture
that it also arises as the scaling limit in various other models above the
upper-critical dimension, such as the incipient infinite lattice tree above 8
dimensions, the incipient infinite cluster for unoriented percolation, uniform
spanning trees above 4 dimensions, and invasion percolation above 6 dimensions.
This paper also serves as a survey of recent results linking super-Brownian to
scaling limits in statistical mechanics.
