Critical two-point functions for long-range statistical-mechanical models in high dimensions. (English) Zbl 1342.60162

This paper deals with self-avoiding walks, percolation and the \(d\)-dimensional Ising model. For \(d\) greater than the upper critical dimension, the authors find the asymptotical form of the critical two-point function. The two-point function indicates how likely the spins located at those two sites point in the same direction for the Ising model. If it decays fast enough to be summable, then there is no macroscopic order. The summability of the two-point function is lost as soon as the model parameter is above the critical point and, therefore, it is naturally hard to investigate critical behavior. The authors overcome those difficulties identifying an asymptotic expression of the critical two-point function for finite-range models, such as the nearest-neighbor model for long-range models, especially when the 1-step distribution for the underlying random walk decays in powers of the distance. As is shown in this paper, the decay rate of the subcritical two-point function is the same as the 1-step distribution of the underlying random walk.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43 Percolation
Full Text: DOI arXiv Euclid


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