M.Sc Student | Levit Anna |
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Subject | Stochastic Geometry of Quantum Currie-Weiss Model |

Department | Department of Industrial Engineering and Management |

Supervisor | Professor Dmitry Ioffe |

The phenomena of phase transition in random fields and percolation models is known to be a topic of utmost importance and has been studied in vast generality. One of the special cases of great interest being the classical Erdős -Rényi model of random graph. In this model, each two vertices, of the complete graph , are connected with probability independently of all other edges, where corresponds to the inverse temperature in the classical mean field Curie-Weiss model. The main result for this model states that when the size of the system tends to infinity phase transition is shown to occur at the critical value . This result corresponds to the phase transition for spontaneous magnetization in the mean field Curie-Weiss model and for long/short range order transition.

The quantum case of Erdős -Rényi model of random graphs incorporates an additional parameter , which represents the power of the field in the Curie-Weiss model in

transversal field. As a result the short and long range transition should be studied in the -quarter plane.

In this work we explicitly compute the corresponding critical curve , which decomposes the -quarter plane into two off-critical regions and , where LRO and SRO stand for the long (respectively short) range order. Our main result states that there is a long range order in the sense that the probability of two points and being connected does not vanish when the size of the system tends to infinity. Contrary to this, such probability vanishes in the limit whenever .

As in the classical case we derive results on the two-point correlation functions and sizes of connected components in both the short and long range order regions. Namely, for with probabilities of order there is a unique giant connected component of size , whereas for all the connected components of have sizes of the order or less. In this way the classical case corresponds to the limiting point on , which corroborates the corresponding existing results in the classical Erdős -Rényi.