Abstract: We give a new construction of the incipient infinite cluster (IIC) associated with high-dimensional percolation in a broad setting and under minimal assumptions. Our arguments differ substantially from earlier constructions of the IIC; we do not directly use the machinery of the lace expansion or similar diagrammatic expansions. We show that the IIC may be constructed by conditioning on the cluster of a vertex being infinite in the supercritical regime 
and then taking 
. Furthermore, at criticality, we show that the IIC may be constructed by conditioning on a connection to an arbitrary distant set 
, generalizing previous constructions where one conditions on a connection to a single distant vertex or the boundary of a large box.
The input to our proof is the asymptotic for the two-point function obtained by Hara, van der Hofstad, and Slade. Our construction thus applies in all dimensions for which those asymptotics are known, rather than explicitly requiring convergence of a diagrammatic expansion. Our construction will be instrumental in upcoming work related to structural properties and scaling limits of various objects involving high-dimensional percolation clusters at and near criticality.
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