Before revision:  Our main result establishes a sharp computational transition at the point \lambda=1 ; in fact, our techniques apply more generally to the whole unit circle|\lambda|=1. We show #P-hardness for approximating the partition function on graphs of maximum degree \Delta when b, the edge-interaction parameter, is in the interval (0,\frac{\Delta-2{\Delta}} and \lambda is a non-real on the unit circle .
After revision:  Our main result establishes a sharp computational transition at the point \lambda=1 , and more generally on the entire unit circle. For an integer \Delta\geq 3 and edge interaction parameter b\in (0,1) we show #P-hardness for approximating the partition function on graphs of maximum degree \Delta when b, the edge-interaction parameter, is in the interval (0,\frac{\Delta-2{\Delta}} and \lambda is a non-real on the unit circle .