  The incipient infinite cluster (IIC) measure is the percolation measure at
criticality conditioned on the cluster of the origin to be infinite. Using the
lace expansion, we construct the IIC measure for high-dimensional percolation
models in three different ways, extending previous work by the second author
and Jarai. We show that each construction yields the same measure, indicating
that the IIC is a robust object. Furthermore, our constructions apply to
spread-out versions of both finite-range and long-range percolation models. We
also obtain estimates on structural properties of the IIC, such as the volume
of the intersection between the IIC and Euclidean balls.
