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Long range order and giant components of quantum random graphs. (English) Zbl 1138.82016

Summary: Mean field quantum random graphs give a natural generalization of classical Erdős-Rényi percolation model on complete graph \(G_N\) with \(p= \beta/N\). Quantum case incorporates an additional parameter \(\lambda\geq0\), 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 \(\gamma_c\).

MSC:

82B43 Percolation
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B10 Quantum equilibrium statistical mechanics (general)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
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