Counting metastable states of Ising spin glasses on arbitrary graphs

Using a field-theoretical representation of the Tanaka-Edwards integral, we develop a method to systematically compute the number Ns of one-spin stable states (local energy minima) of a glassy Ising system with nearest-neighbor interactions and random Gaussian couplings on an arbitrary graph. In particular, we use this method to determine Ns for K -regular random graphs and d -dimensional regular lattices for d=2,3 . The method works also for other graphs. Excellent accuracy of the results allows us to observe that the number of local energy minima depends mainly on local properties of the graph on which the spin glass is defined.