Ising model with spins S=1/2 and 1 on directed and undirected Erd'os-R\'enyi random graphs

摘要

Using Monte Carlo simulations we study the Ising model with spin S=1/2 and 1on {\it directed} and {\it undirected} Erd\"os-R\'enyi (ER) random graphs, with$z$ neighbors for each spin. In the case with spin S=1/2, the {\it undirected}and {\it directed} ER graphs present a spontaneous magnetization in theuniversality class of mean field theory, where in both {\it directed} and {\itundirected} ER graphs the model presents a spontaneous magnetization at $p =z/N$ ($z=2, 3, ...,N$), but no spontaneous magnetization at $p = 1/N$ which isthe percolation threshold. For both {\it directed} and {\it undirected} ERgraphs with spin S=1 we find a first-order phase transition for z=4 and 9neighbors.