An effective lattice theory for Polyakov loops

We derive an effective action for SU(2) Polyakov loops using inverse Monte Carlo techniques. The couplings of the effective theory are extracted by combining two independent approaches. First, we employ novel lattice Schwinger--Dyson equations which reflect the SU(2) x SU(2) invariance of the functional Haar measure. Second, we require that the effective ensemble reproduces the single--site distribution of the Polyakov loops. The latter is flat below the critical temperature implying that the (untraced) Polyakov loop is distributed uniformly over its target space, the SU(2) group manifold. This allows for an analytic determination of the Binder cumulant and the distribution of the mean--field, which turns out to be approximately Gaussian