Self-avoiding walks on finite graphs of large girth

We consider self-avoiding walk on finite graphs with large girth. We study a few aspects of the model originally considered by Lawler, Schramm and Werner on finite balls in Z^d. The expected length of a random self avoiding path is considered. We also define a "critical exponent" $\gamma$ for sequences of graphs of size tending to infinity, and show that $\gamma = 1$ in the large girth case.