Long Range Order and Giant Components of Quantum Random Graphs
2007, v.13, №3, 469-492
Mean field quantum random graphs give a natural generalization of classical Erdos - Renyi percolation model on complete graph $G_N$ with $p =\beta /N$. Quantum case incorporates an additional parameter $\lambda\geq 0$, and the short-long range order transition should be studied in the $(\beta ,\lambda)$-quarter plane. In this work we explicitly compute the corresponding critical curve $\gamma_c$, and derive results on two-point functions and sizes of connected components in both short and long range order regions. In this way the classical case corresponds to the limiting point $(\beta_c ,0) = (1,0)$ on $\gamma_c$.
Keywords: quantum Curie - Weiss model,FK representation,percolation,giant components of random graphs,branching random walks