Long range frustration in finite connectivity spin glasses: A mean field theory and its application to the random $K$-satisfiability problem

Shortened abstract: A mean field theory of long range frustration is constructed for spin glass systems with quenched randomness of vertex--vertex connections and of spin--spin coupling strengths. This theory is applied to a spin glass model of the random $K$-satisfiability problem (K=2 or K=3). The zero--temperature phase diagram of the $\pm J$ Viana--Bray model is also determined, which is identical to that of the random 2-SAT problem. The predicted phase transition between a non-frustrated and a long--rangely frustrated spin glass phase might also be observable in real materials at a finite temperature.