  We study the variable-length ensemble of self-avoiding walks on the complete
graph. We obtain the leading order asymptotics of the mean and variance of the
walk length, as the number of vertices goes to infinity. Central limit theorems
for the walk length are also established, in various regimes of fugacity.
Particular attention is given to sequences of fugacities that converge to the
critical point, and the effect of the rate of convergence of these fugacity
sequences on the limiting walk length is studied in detail. Physically, this
corresponds to studying the asymptotic walk length on a general class of
pseudocritical points.
