The asymptotic behaviour of symbolic generic initial systems of generic points

Consider the ideal I⊆K[x,y,z] corresponding to points p1,…,pr of P2. We study the symbolic generic initial system {gin(I(m))}m of such an ideal and its behaviour as m gets large. In particular, we describe the limiting shape of this system explicitly when p1,…,pr are generic, assuming that the Uniform SHGH Conjecture holds for r≥9. The symbolic generic initial system and its limiting shape reflect information about the asymptotic behaviour of Hilbert functions of fat point ideals.