## Abstract

We propose the finite-size scaling of correlation functions in finite systems near their critical points. At a distance r in a d-dimensional finite system of size L, the correlation function can be written as the product of |r|^{−(d−2+η)} and a finite-size scaling function of the variables r/L and tL^{1/v}, where t = (T − T_{c})=T_{c}, η is the critical exponent of correlation function, and v is the critical exponent of correlation length. The correlation function only has a sigificant directional dependence when |r| is compariable to L. We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations. We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponent η.

Original language | English |
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Article number | 120511 |

Journal | Science China: Physics, Mechanics and Astronomy |

Volume | 61 |

Issue number | 12 |

DOIs | |

State | Published - 1 Dec 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2018, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.

### Funding

This work was supported by the Key Research Program of Frontier Sciences, Chinese Academy of Sciences (Grant No. QYZD-SSW-SYS019). In addition, YongWen Zhang received a postdoctoral fellowship funded by the Kunming

Funders | Funder number |
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Key Research Program of Frontier Sciences | |

Chinese Academy of Sciences | QYZD-SSW-SYS019 |

## Keywords

- correlation function
- critical phenomena
- finite-size scaling
- lattice model