Correction-to-scaling exponent for percolation and the Fortuin-Kasteleyn Potts model in two dimensions Articles uri icon

publication date

  • March 2025

start page

  • 034108 -1

end page

  • 034108-13

issue

  • 3

volume

  • 111

International Standard Serial Number (ISSN)

  • 1539-3755

Electronic International Standard Serial Number (EISSN)

  • 1550-2376

abstract

  • The number ns of clusters (per site) of size s, a central quantity in percolation theory, displays
    at criticality an algebraic scaling behavior of the form ns ≃ s−τ A(1 + Bs−Ω). For the Fortuin–
    Kasteleyn representation of the Q-state Potts model in two dimensions, the Fisher exponent τ is
    known as a function of the real parameter 0 ≤ Q ≤ 4, and, for bond percolation (the Q → 1
    limit), the correction-to-scaling exponent is derived as Ω = 72/91. We derive theoretically the
    exact formula for the correction-to-scaling exponent Ω = 8/[(2g + 1)(2g + 3)] as a function of the
    Coulomb-gas coupling strength g, which is related to Q by Q = 2 + 2 cos(2πg). Using an efficient
    Monte Carlo cluster algorithm, we study the O(n) loop model on the hexagonal lattice, which is in
    the same universality class as the Q = n2 Potts model, and has significantly suppressed finite-size corrections and critical slowing-down. The predictions of the above formula include the exact value for percolation as a special case and agree well with the numerical estimates of Ω for both the critical and tricritical branches of the Potts model.

subjects

  • Mathematics

keywords

  • classical statistical mechanics; critical phenomena; classical spin models; equilibrium lattice models; monte carlo methods