Statistical physics of loopy interactions: Independent-loop approximation and beyond

We consider an interacting system of spin variables on a loopy interaction graph, identified by a tree graph and a set of loopy interactions. We start from a high-temperature expansion for loopy interactions represented by a sum of non-negative contributions from all the possible frustration-free loop configurations. We then compute the loop corrections using different approximations for the nonlocal loop interactions induced by the spin correlations in the tree graph. For distant loopy interactions, we can exploit the exponential decay of correlations in the tree interaction graph to compute loop corrections within an independent-loop approximation. Higher orders of the approximation are obtained by considering the correlations between the nearby loopy interactions involving larger number of spin variables. In particular, the sum over the loop configurations can be computed "exactly" by the belief propagation algorithm in the low orders of the approximation as long as the loopy interactions have a tree structure. These results might be useful in developing more accurate and convergent message-passing algorithms exploiting the structure of loopy interactions.