Rigorous decimation-based construction of ground pure states for spin glass models on random lattices

A constructive scheme for determining pure states (clusters) at very low temperature in the 3-spins glass model on a random lattice is provided, in full agreement with Parisi's one step replica symmetry breaking (RSB) scheme. Proof is based on the analysis of an exact decimation procedure. When the number c of couplings per spin is smaller than some critical value c_d, all spins are eliminated at the end of decimation (RS phase). In the range c_d<c<c_s, a reduced Hamiltonian is left; each ground state (GS) of the latter is a "seed" from which a cluster of GS of the original Hamiltonian can be reconstructed. Above c_s, GS are frustrated with an energy per spin larger than -c. The number of GS in each cluster, the number of clusters, the distances between GS are calculated and correspond to RSB predictions.