We study several kinds of large critical percolation clusters
in two dimensions. We show that from the microscopic (lattice
scale) perspective these clusters can be described by Kesten's
incipient infinite cluster (IIC), as was conjectured by
Aizenman. More specifically, we establish this for incipient spanning
clusters, large clusters in a finite box and the inhomogeneous
model of Chayes, Chayes and Durrett. Our results prove the
equivalence of several natural definitions of the IIC.
¶ We also show that for any $k \ge 1$ the difference in size between
the $k$th and $(k+1)$st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes
to infinity. In addition, the distribution of the
Chayes--Chayes--Durrett
cluster is shown
to be singular with respect to the IIC.