Limit laws for UGROW random graphs

Representations are found for a limit law L(Z(k,p))L(Z(k,p)) obtained from an expanding sequence of random forests containing nn nodes with p∈(0,1]p∈(0,1] a probability controlling bond formation. One implies that Z(k,p)Z(k,p) is stochastically decreasing as kk increases and that norming gives an exponential limit law. Limit theorems are given for the order of component trees. The proofs exploit properties of the gamma function.