Linking of random p-spheres in Z(d)

We consider the number of embeddings of k p-spheres in Z(d), with p + 2 les s than or equal to d less than or equal to 2p + 1, stratified by the p-dime nsional volumes of the spheres. We show for p + 2 < d that the number of em beddings of a fixed link type of k equivolume p-spheres grows with increasi ng p-dimensional volume at an exponential rate which is independent of the link type. For d = p + 2 we derive similar results both for links of unknot ted p-spheres and for "augmented" links where each component p-sphere can h ave any knot type, and similar but weaker results when the spheres are of s pecified knot type.