In 1983, Pansu showed that the asymptotic cone of a finitely generated nilpotent group is a nilpotent Lie group with a left-invariant Carnot-Carthéodory metric. In this talk, we will explore this class of metrics on the real Heisenberg group which appear as the asymptotic cones of the integer Heisenberg group with different generating sets. In particular, we will study their horofunction boundaries and discuss how this boundary can help us study random walks in the Heisenberg group. This is joint work with Sebastiano Nicolussi Golo.