The crossover region between long-range and short-range interactions for the critical exponents.

*(English)*Zbl 1318.82017It is well known that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. There is a value of the exponent that separates the short-range behavior from the long-range behavior. The following natural question is interesting: What happens at this crossover point? In this paper, the authors propose a general form for the crossover function. Namely, they find that there is a non-trivial behavior at the crossover point, i.e., one has logarithmic correlations to the standard power law behavior. They compare the obtained predictions with the results of numerical simulations for two-dimensional long-range percolation.

Reviewer: Farruh Mukhamedov (Kuantan)

##### MSC:

82B27 | Critical phenomena in equilibrium statistical mechanics |

82B43 | Percolation |

82D40 | Statistical mechanics of magnetic materials |

05C83 | Graph minors |

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\textit{E. Brezin} et al., J. Stat. Phys. 157, No. 4--5, 855--868 (2014; Zbl 1318.82017)

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##### References:

[1] | Sak, J, Recursion relations and fixed points for ferromagnets with long-range interactions, Phys. Rev. B, 8, 281, (1973) |

[2] | Sak, J, Low-temperature renormalization group for ferromagnets with long-range interactions, Phys. Rev. B, 15, 4344, (1977) |

[3] | Fisher, ME; Ma, SK; Nickel, BG, Critical exponents for long-range interactions, Phys. Rev. Lett., 29, 917, (1972) |

[4] | Luijten, E; Blöte, HWJ, Boundary between long-range and short-range critical behavior in systems with algebraic interactions, Phys. Rev. Lett., 89, 025703, (2002) |

[5] | M. Picco, preprint arXiv:1207.1018 (2012). |

[6] | Blanchard, T; Picco, M; Rajabpour, MA, Influence of long-range interactions on the critical behavior of the Ising model, Europhys. Lett., 101, 56003, (2013) |

[7] | Angelini, MC; Parisi, G; Ricci-Tersenghi, F, Relations between short-range and long-range Ising models, Phys. Rev. E, 89, 062120, (2014) |

[8] | Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Oxford Univ. Press, Oxford (1996) · Zbl 0865.00014 |

[9] | Ma, S.K.: Critical exponents for charged and neutral bose gases above \(λ \) points. Phys. Rev. Lett. 29, 1311 (1972) |

[10] | Abe, R; Hikami, S, Critical exponents and scaling relations in 1/n expansion, Prog. Theor. Phys., 49, 442, (1973) |

[11] | Barone, L.M., Marinari, E., Organtini, G., Ricci-Tersenghi, F.: Scientific Programming. World Scientific, Singapore (2013) · Zbl 1283.68001 |

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