[74.1.2.1] This section discusses universal amplitudes such as those defined in (1.6) and their ratios. [74.1.2.2] In numerical simulations of critical phenomena amplitude ratios such as (1.7) are used routinely to extract critical parameters and exponents from simulations of finite systems. [74.1.2.3] It is then of interest to analyze finite size amplitude ratios within the present framework.

[74.1.3.1] The absolute moment of order for the ensemble averages of in a finite and noncritical system is found from equations (4.4) and (3.16) as

(7.1) | |||

(7.2) |

where the amplitude of the finite, discrete and noncritical system is given as

(7.3) |

and the function is defined from equation (4.7) by replacing with and extracting a factor . [74.2.0.1] In the ensemble limit one obtains from this and (3.17) the result

(7.4) |

for the critical ensemble scaling amplitude of order in an infinite system. [74.2.0.2] The subscript is again a reminder for the ensemble scaling limit. [74.2.0.3] The integral in (7.4) can be evaluated for as

(7.5) |

which is valid for and . [74.2.0.4] A derivaton of this result is given in Appendix B. [74.2.0.5] This allows to calculate the general moment ratios

(7.6) |

with in the ensemble limit. [74.2.0.6] Figure 6 shows a twodimensional plot of the ratio .

[74.2.1.1] If equation (7.5) is used to analytically continue beyond the regime the traditional fourth order cumulant for the order parameter is found to exhibit special problems if . [74.2.1.2] This is mainly due to the presence of the factor in (7.5). [74.2.1.3] The divergence must somehow become absorbed by the cutoff factor in the finite size scaling limit. [74.2.1.4] Assuming that this is indeed the case it is then of interest to consider the quantity

(7.7) |

[page 75, §0] in the finite size scaling limit assuming that it exists. [75.1.0.1] Then the traditional finite size cumulant becomes

(7.8) |

[75.1.0.2] The interest in this formal expression is that it is still singular. [75.1.0.3] Within the domain it has simple poles along the lines

(7.9) |

and zeros along the lines

(7.10) |

[75.1.0.4] For the traditionally studied order parameter cumulant, i.e. setting , the pole at implies a divergence whenever , i.e. in mean field theory. [75.1.0.5] This result is consistent with the divergence found in conformal field theory [17]. [75.1.0.6] Note that the points along the singular mean field line are intersection points with a line of zeros.

[75.1.1.1] Irrespective of these problems it is of interest to estimate values for the traditional order parameter cumulant ratio because much previous work has focussed on it. [75.1.1.2] Within the present approach this is possible from the knowledge of the scaling functions if it is assumed that the identification of with periodic boundary conditions holds generally. [75.1.1.3] If the scaling functions with in Figures 2 through 4 are simply truncated sharply at , and subseqently rescaled to unit norm and variance, the order parameter cumulant may be calculated as usual, and it will depend upon the nonuniversal cutoff at . [75.2.0.1] The results of such a cutoff procedure are displayed in Figure 7 for the cases . [75.2.0.2] It is seen that the cumulant is distinctly cutoff dependent. [75.2.0.3] Note that all curves appear to diverge as the cutoff increases.

[75.2.0.4] For the cases and some structure appears between and corresponding to the strong curvature in this region seen in Figures 2 and 3. [75.2.0.5] For the -Ising case the curve is flat up to about twice the maximal value for the simulations of Bruce and coworkers [43, 45]. [75.2.0.6] Figure 5 provides a possible explanation for the poor agreement between the value observed in simulations of the fivedimensional Ising model [11, 46] and the mean field calculation from [15]. [75.2.0.7] The simulation result is indicated as the solid arrow, the analytical result as the dashed arrow pointing to the curve . [75.2.0.8] The small difference in the cutoff corresponding to these values suggests that the discrepancy may result from different nonuniversal (but most likely smooth) cutoffs in the two estimates.

[75.2.1.1] Finally, the fact that the value of the universal shape parameter appears to be related to the choice of bondary conditions suggests a method of constructing critical amplitude ratios which do not depend on boundary conditions, or other factors influencing . [75.2.1.2] The basic idea is to use the difference of two independent observations of ensemble averages or sums. [75.2.1.3] Let and be two independent measurements and their difference. [75.2.1.4] The limiting distribution function for and at criticality is given in equation (4.2). [75.2.1.5] Then the difference has the distribution function

(7.11) |

[page 76, §0] in which the width is doubled, but has disappeared. [76.1.0.1] The fractional difference moment ratio is formed analogously to the moment ratio as

(7.12) |

and it has a universal value depending only on the scaling dimension of as long as . [76.1.0.2] If the scaling dimension is universal then the fractional difference moment ratio is independent of boundary conditions. [76.1.0.3] Plotting as a function of length scale and temperature should then allow to extract the critical exponent.