Mean-field theory of spin-glasses with finite coordination number.

The mean-field theory of dilute spin-glasses is studied in the limit where the average coordination number is finite. The zero-temperature phase diagram is calculated and the relationship between the spin-glass phase and the percolation transition is discussed. The present formalism is applicable also to graph optimization problems. The mean-field theory (MFT) of spin-glasses usually refers to an infinite range-system of N spins; each one of them is connected to the remaining N — 1 spins. ' In this paper, the MFT of dilute spin-glasses is studied. If, after dilution of the bonds, the average number of bonds per spin remains of O(N), the dilution does not affect the behavior of the system. If, however, the average coordination number is finite, some new physics is expected to emerge. In particular there will be an interesting interplay between the statistical-mechanical frustration and the geometric connectivity fluctuations. Also, one might expect that some features of the diluted system are closer to the realm of the short-range spin-glass (SG) system. Besides the relevance to the low-temperature properties of spin-glasses, the theory of dilute spin-glasses has important applications in graph optimization problems. Some of these problems can be mapped into random, frustrated Ising models with highly diluted infinite-range interactions. For these problems, mean-field theory should yield exact results in the thermodynamic limit. The MFT of dilute SG's has been previously systematically studied only in the neighborhood of the transition temperature. However, some of the interesting properties of the system are revealed at low temperatures. In fact, earlier treatments of the low-T phase failed to a. "-rive at a consistent theory which incorporates the all-important frustration of the system. We present here a where S; =+'1 (i =1,. .. , N), and the J;I's are infi nite-ranged random interactions. Their probability distribution is It describes a network of bonds which is highly diluted: The average coordination number of each spin is c which is taken to be on the order of 1. The distribution of the surviving bonds is given by f(J~)which is norm. alized to unity. Because the average number of bonds is cN/2 (and not N /2) the scale of J;I must be on the order of 1 to achieve the appropriate thermodynamic limit. Since there are no length scales in the problem, a mean-field theory is expected to give an exact description of the system in the thermodynamic limit (N ~). Indeed, using …