The K-sat problem in a simple limit

The statistical mechanics of the random K-satisfiability (K-SAT) problem has been the object of many studies in the last years [1][2][3]. The K-SAT was the first problem to be shown to be Nondeterministic Polynomial (NP) complete [4] and the importance of such a model is that it provides a prototype for all the NP complete problems in complexity theory of computer science as well as in statistical mechanics of disordered and glassy systems, in computational biology and in other fields. The NP complete are those problems whose solution, or the certainty that they have no solutions, can only be found (in the worst case) by algorithms with a running time of computation that likely grows faster than polynomially (namely exponentially) with the number of variables N of the system. The K-SAT model also allows a basic understanding of the connection between the SAT/UN-SAT phase transition in random combinatorial structures [1][2] and the transition between a Replica Symmetric structure and a structure where the symmetry is broken in the frame of spin-glasses [5]. Yet the question stays open of how exactly the typical-case complexity theory of computer science and the Replica Symmetry Breaking (RSB) transition are related. One of the main aim of the recent research on this model has been to understand better the structure of solutions, above all at the borderline between the region of diagram of phase where the