EXACT FREE-ENERGY FUNCTIONALS FOR NON-SIMPLY-CONNECTED LATTICES

We consider the nearest-neighbor Ising model in thermal equilibrium on a network with no required regularity or symmetry properties. Both co upling strengths and external fields are site-dependent. The objective is to describe this system in terms of a free energy magnetization fu nctional whose conjugate variables are the external fields. For simply connected networks, this inverse problem has a local structure. On ge neralizing to loops, the local structure remains if the description is expanded in an overcomplete fashion to include a collective amplitude with respect to which the free energy is stationary. For more complex connectivity, a superbond representation is developed in terms of whi ch the system can be described by a combined auxiliary set of branch a nd node collective variables.