  We analyze models of genome evolution based on both restricted and
unrestricted double-cut-and-join (DCJ) operations. We compare the number of
operations along the evolutionary trajectory to the DCJ distance of the genome
from its ancestor at each step, and determine at what point they diverge: the
process escapes from parsimony. Adapting the method developed by Berestycki and
Durret (2006), we estimate the number of cycles in the breakpoint graph of a
random genome at time $t$ and its ancestral genome by the number of tree
components of an Erd\"os-R\'enyi random graph constructed from the model of
evolution. In both models, the process on a genome of size $n$ is bound to its
parsimonious estimate up to $t\approx n/2$ steps.
