A complexity trichotomy for k-regular asymmetric spin systems with complex edge functions

We prove a complexity trichotomy theorem for a class of partition functions over k-regular graphs, where the signature is complex valued and not necessarily symmetric. In details, we establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: For every parameter setting in $C$ for the spin system, the partition function is either (1) computable in polynomial time for every graph, or (2) #P-hard for general graphs but computable in polynomial time for planar graphs, or (3) #P-hard even for planar graphs.