A Monte Carlo Study To The Geometric Property Of Percolation In High Dimensions

In nature,both the transition from gas to liquid or the transition from ferromagnet to paramagnetic magnetism,the transition from one phase to another is everywhere.The study of phase transition not only deepens people's awareness of the laws of na-ture,but also promotes the development of modem materials science.Scientists have put forward the theory of phase transition,in which Landau's theory of phase transition and renormalization group theory and so have made significant progress.However,for many model theory calculations are so complex that it is difficult to complete,peo-ple thus have introduced Monte Carlo methods which is by means of a computer with powerful computing power.In recent years,the Monte Carlo method has played a sig-nificant role in the study of phase transition.In this paper,the Monte Carlo method is used to study one of the most simple classical statistical model:percolation model.In this paper,we first introduce some backgrounds of the phase transition phe-nomena and the Monte Carlo method.Then we give some elaborations on the phys-ical background of the two percolation mean field models:the Bethe lattice and the complete graph.Then we introduce the geometric properties of high dimensional per-colation model based on Monte Carlo method.In this paper,we make three kinds of geometric classification on the critical percolation on complete graph and in seven di-mension to compare and analyze the difference between the physical quantities,and find that the prediction of some critical exponents of mean field model is not hold on perco-lation above upper critical dimension,and the similarities and differences between the geometric properties of the two models are systematically revealed.Finally,we make a summary of the content of this article,and make a prospect for future research.