Some Rigorous Results for the Diluted Multi-species SK Model

We consider the diluted multi-species Sherrington–Kirkpatrick (DMSK) model in which the variance of disorders depend on the species the particles belong to, and the number of edges within each block is diluted. First, we find the annealed region of the DMSK model at high temperature and compute the corresponding free energy. Next, we get a fluctuation result for the overlap vector through a differential method. Lastly, by using cavity method, we obtain the corresponding replica symmetric bound and r-step of replica symmetry breaking bound.