Dynamical mean-field theory as a random loop problem

In dynamical mean-field theory (DMFT) the Anderson lattice model is mapped onto the impurity model, with the density of states determined from a self- consistence condition (scc). The mapping is rigorous in infinite spatial di mensions d. It can be diagrammatically modelled as self-avoiding loops. Whi le at finite d > 4 the number of mathematical self-avoiding loops is neglig ible compared to all random loops, to the mathematical sec at infinite d th ey contribute a fraction 1/e to all random loops. The limits of d --> infin ity and infinite loop length cannot be interchanged, thus making 1/d-correc tions to the sec questionable. We find the analogous result for the DMFT lo op. We also discuss numerical difficulties arising in the infinite-U limit of the Anderson lattice model, and analytically simulate them.