[69.1.1.1] We consider the two-dimensional Ising model on a square lattice of side length . The spins interact according to the Hamiltonian where is the ferromagnetic coupling strength and is an external field. [69.1.1.2] The first summation runs over all nearest neighbour pairs on the lattice. [69.1.1.3] The order parameter is the magnetization per spin

(1) |

whose value fulfills . [69.1.1.4] In the following we set and also the Boltzmann constant to unity. [69.1.1.5] We denote the temperature by , and write for the magnetic field.

In this paper we focus on the probability density of the order parameter defined as

(2) |

where , is a Kronecker , and where is such that is an integer not larger than . [69.1.1.6] The probability density depends parametrically on temperature , field and system size ,

(3) |

It is also called order parameter distribution. [69.1.1.7] In the following we limit ourselves to the case , and hence . [69.1.1.8] The critical order parameter distribution is obtained in the limit and where is the critical temperature. [69.1.1.9] There are different ways of taking this limit (see [23] for an overview). [69.1.1.10] Traditionally this limit is understood as the finite-size scaling limit defined by

(4) |

where is the temperature dependent spin-spin-correlation length for the infinite system. [69.1.1.11] Note that in an infinite system as . [69.1.1.12] A second way to take the limit is the finite ensemble scaling limit defined through [23]

(5) |

[69.2.0.1] All other possibilities for taking the limits are discussed in [23]. [69.2.0.2] It is often postulated that fulfills the finite size scaling hypothesis

(6) |

in the finite-size scaling limit. [69.2.0.3] Here is the reduced temperature, is a universal scaling function and is the correlation length exponent. For the two-dimensional Ising model .

[69.2.1.1] The traditional finite-size scaling hypothesis [13, 24] for the critical order parameter distribution assumes that

(7) |

where is the universal scaling function of the order parameter distribution and is the order parameter exponent. [69.2.1.2] For the two-dimensional Ising model [25]

(8) |

Our scaling variable is then

(9) |

Using the scaling assumption (7) one obtains the absolute moments of the critical order parameter distribution

(10) |

where

(11) |

[69.2.1.3] From these one calculates the so called renormalized coupling constant or the Binder cumulant

(12) |

which are often used in studies of critical behaviour because they are independent of at criticality, if all the assumptions are valid.