Finiteness of outer automorphism groups of random right-angled Artin groups

We consider the outer automorphism group Out (A(Gamma)) of the right- angled Artin group A(Gamma) of a random graph Gamma on n vertices in the Erdos-Renyi model. We show that the functions n(-1) (log(n) + log(log(n))) and 1 - n(-1)(log(n) + log(log(n))) bound the range of edge probability functions for which Out(A(Gamma)) is finite: if the probability of an edge in Gamma is strictly between these functions as n grows, then asymptotically Out(A(Gamma)) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out(A(Gamma)) is almost surely infinite. This sharpens a result of Ruth Charney and Michael Farber.