IV Results

A Results for moments and cumulants

[71.1.3.1] In Figure 1 we compare the Binder cumulant for periodic and fixed boundary conditions. [71.1.3.2] We notice that for fixed boundary conditions the values lie generally above those for the periodic case. [71.1.3.3] They are very close to their upper limit $U_{L}=2/3$ , that is expected for a nonvanishing first moment.

Figure 1: Cumulant $U_{L}$ versus $1/L$ for fixed and periodic boundary conditions. Individual curves correspond to $T=2.21$ , $2.22$ , $2.23$ , $2.24$ , $2.25$ , $2.26$ , $2.2691$ , $2.27$ , $2.28$ , $2.29$ , $2.3$ from top (low $T$ ) to bottom (high $T$ ). The curves for $T_{c}$ are marked with a special symbol ( $+$ for periodic and $\circ$ for fixed boundary conditions).

B Results for maxima

[71.1.4.1] For fixed boundary conditions the function $p(m;T,L)$ has a single maximum as a function of $m$ for all values of $T$ and $L$ . For periodic boundary conditions on the other hand there exists a temperature $T^{*}(L)$ above which the distribution has a single maximum, below which it has two local maxima. [71.1.4.2] For $L\to\infty$ the most probable magnetization approaches the exact infinite volume magnetization given by

$m_{{\rm ex}}(T)=\begin{cases}\pm\left(1-\left(\sinh(2/T)\right)^{{-4}}\right)^{{1/8}}&\text{for $T<T_{c}$}\\ 0&\text{for $T\geq T_{c}$}.\end{cases}$

(18)

[71.1.4.3] Therefore we utilize the difference

$\Delta(T,L)=m_{{\rm mp}}(T,L)-m_{{\rm ex}}(T)$

(19)

as a measure for the convergence to the infinite volume result. [71.1.4.4] Here $m_{{\rm mp}}$ is the most probable magnetization defined earlier.

[71.1.5.1] In Figure 2 we plot this difference as a function of temperature for system sizes $L=16,32,64$ for the case of fixed boundary conditions. [71.1.5.2] One sees that for $T\ll T_{c}$ the difference is small. [71.1.5.3] For $T\gg T_{c}$ the difference is rather large. [71.1.5.4] This is surprising, as we shall see in more detail below. [71.1.5.5] One also sees a pronounced maximum around $T_{c}$ . [71.1.5.6] The height of the maximum is so large that the results are clearly not $L$ -converged around $T_{c}$ . [71.1.5.7] The asymptotic value for the maximum value at $T=T_{c}$ is $m=0$ for all boundary conditions. [71.2.0.1] There is no reason to believe that the shape of the critical order parameter distribution has reached its asymptotic limit, if its peak (maximum value) has not reached its asymptotic limit.

Figure 2: Difference $\Delta(T,L)$ defined in (19) between the most probable magnetization $m_{{\rm mp}}(T,L)$ and the exact infinite volume magnetization $m_{{\rm ex}}(T)$ versus temperature for fixed boundary conditions. Different curves correspond to system sizes $L=16(\times)$ , $L=32(\circ)$ and $L=64(\triangle)$ . The vertical dashed line marks $T_{c}$ .

[71.2.1.1] In Figure 3 we plot $\Delta(T,L)$ for the case of periodic boundary conditions. [71.2.1.2] In this case there are two local maxima below $T_{c}$ and in the critical region. [71.2.1.3] Hence there are two curves. [71.2.1.4] Compared to the case of fixed boundary conditions the deviations above $T_{c}$ appear to be smaller. [71.2.1.5] A more detailed comparison to Gaussian behaviour, to be performed below, shows that the deviations above $T_{c}$ are comparable to those in the fixed case.

Figure 3: Difference $\Delta(T,L)$ defined in (19) between the most probable magnetization $m_{{\rm mp}}(T,L)$ and the exact infinite volume magnetization $m_{{\rm ex}}(T)$ versus temperature for periodic boundary conditions. Different curves correspond to system sizes $L=16(\times)$ , $L=32(\circ)$ and $L=64(\triangle)$ . The vertical dashed line marks $T_{c}$ .

C Results for tails

[71.2.2.1] Here we analyze the tails of $p(m)$ . First we find numbers $A,B$ such that the rescaled function

$p_{0}(x)=Ap(B(m-C))$

(20)

where $x=B(m-C)$ and $p_{0}=Ap$ has mean zero, unit norm and unit variance. [71.2.2.2] To facilitate the comparison between periodic and fixed boundary conditions the data for periodic boundary conditions were treated somewhat differently than it is normally done. [71.2.2.3] In the periodic case eq. (20) is applied not to $p(m)$ itself but only to its right half, i.e. the data for $m>0$ . In Figure 4 we show the rescaled functions $p_{0}(x)$ at criticality $T=T_{c}$ for fixed and periodic boundary conditions. [71.2.2.4] The data collapse at $T_{c}$ is found to be generally good.

Figure 4: Rescaled order parameter distributions $p_{0}(x)$ for $T=T_{c}$ and $L=16,32,64$ for fixed and periodic boundary conditions. (Only the right half of the distribution is scaled and shown for periodic boundary conditions).

[71.2.3.1] To analyze the tails we split the function $p_{0}(x)$ at the peak into the left and right tail. More precisely we find the functions

$\begin{array}[]{r}{p_{0}}_{r}(x)=p_{0}(x-x_{{\rm peak}})\qquad\text{for $x>x_{{\rm peak}}$}\\ {p_{0}}_{l}(x)=p_{0}(x_{{\rm peak}}-x)\qquad\text{for $x<x_{{\rm peak}}$}\end{array}$

(21)

where $x_{{\rm peak}}$ is the position of the maximum. [71.2.3.2] To exhibit stretched exponential tails we calculate the functions

$q(y)=\frac{{\rm d}\log _{{10}}(-\log _{{10}}{p_{0}}_{i})}{{\rm d}\log _{{10}}x}$

(22)

where $i=l,r$ and plot them against $y=\log _{{10}}x$ . [71.2.3.3] In this way of plotting the data a tail of the form $p_{0}(x)\sim B(x+c)^{\beta}\exp(-A(x+c)^{\alpha})$ corresponds to the function

$q(y)=\alpha\left(1-\frac{cf_{A}-f_{B}+(\beta 10^{y})/(\alpha(10^{y}+c))}{(10^{y}+c)f_{A}-f_{B}}\right)$

(23)

where $f_{A}=A(10^{y}+c)^{{\alpha-1}}$ and $f_{B}=\ln B+\beta\ln(10^{y}+c)$ . [71.2.3.4] The exponent $\alpha$ can be easily identified as a plateau at the value $\alpha$ . In these plots the far tail regime corresponds to large values of $x$ . [71.2.3.5] A standard normal (Gaussian) distribution $(1/\sqrt{2\pi})\exp(-x^{2}/2)$ corresponds to the function

$g(y)=2\left(1-\frac{\log _{{10}}(2\pi)}{\log _{{10}}({\rm e})10^{{2y}}+\log _{{10}}(2\pi)}\right)$

(24)

[page 72, §0] where $y=\log _{{10}}x$ . [72.1.0.1] We note that our choice to split $p_{0}(x)$ at the peaks is natural, and we believe, the only reasonable choice away from $T_{c}$ . [72.1.0.2] At $T_{c}$ , multiplication of algebraic prefactors or splitting the distribution differently into right and left tails does not affect the results of the following analysis of the far tail region.

[72.1.1.1] In the following we plot the results of our tail analysis for three selected temperatures $T=1.5$ , $T=2.2691\approx T_{c}$ and $T=3.5$ . [72.1.1.2] We have chosen these temperatures to demonstrate the degree of convergence with respect to $L$ below $T_{c}$ , at $T_{c}$ and above $T_{c}$ . [72.1.1.3] Below and above $T_{c}$ the central limit theorem predicts Gaussian tails which would correspond to a plateau at 2 in our plots. [72.1.1.4] At $T_{c}$ theory predicts a right tail of the form $x^{7}\exp(-x^{{16}})$ corresponding to a plateau at 16.

[72.1.2.1] In Figures 5-7 these are presented for three different temperatures. [72.1.2.2] Each of the three figures shows the left tail in the upper row and the right tail in the lower row of the figure. [72.1.2.3] Fixed boundary conditions appear in the left column, and periodic boundary conditions in the right column.

Figure 5: Tail analysis for $T=3.5$ . The solid lines represent the standard normal (Gaussian) distribution given in eq. (24).

[72.1.3.1] All the plots in Figure 5 for the high temperature $T=3.5$ are similar. [72.1.3.2] All tails approach the solid line from the top as $L$ is increased. [72.1.3.3] The solid line represents a Gaussian distribution as expected from the central limit theorem. [72.1.3.4] However, all the data, even those for $L=64$ , match the Gaussian form over a relatively narrow range in $\log _{{10}}x$ . [72.1.3.5] Note that this match still means agreement over many orders of magnitudes in the probability. [72.1.3.6] In our analysis the plateau at 2 is beginning to become visible in the data. [72.1.3.7] The emergence of the Gaussian tails $\sim\exp(-x^{2}/2)$ is slow, and system sizes of at least $L\approx 256$ are necessary to clearly show the plateau at 2.

Figure 6: Tail analysis for $T=T_{c}$ . The solid lines in the two upper figures represent the standard normal (Gaussian) distribution given in eq. (24). In the lower figures the solid lines are guides to the eye based on fitting the far right tails with $p_{0}(x)=a((x+c)/b)^{7}\exp(-((x+c)/b)^{{16}})$ where $a=1.31$ , $b=8.59$ , $c=7.79$ for fixed and $a=1.45$ , $b=4.59$ , $c=3.69$ for periodic boundary conditions.

[72.1.4.1] Figure 6 shows the results for the tails of the critical order parameter distribution. [72.1.4.2] The curves for the left tails (upper row) for all the three different system sizes nearly collapse. [72.1.4.3] Deviations appear only at larger values of the scaling variable $x$ . [72.1.4.4] The data collapse makes it difficult to see any systematic approach to the limiting function for infinite systems. [72.1.4.5] It should be kept in mind that the scaling function has not reached its form for infinite systems, because the data collapse extends only over a narrow absolute range in the scaling variable $x$ . [72.1.4.6] For fixed boundary conditions a plateau seems to develop at around $0.75$ . [72.1.4.7] It would correspond to an anomalous stretched exponential tail of the scaling function. [72.1.4.8] To the best of our knowledge this has not been observed or predicted up to now.

[72.1.5.1] Next we turn to the right tail of the order parameter distribution at $T_{c}$ . [72.1.5.2] This tail is expected to behave as $\sim x^{7}\exp(-x^{{16}})$ corresponding to a plateau at 16 for large $x$ [4, 3]. [72.1.5.3] Our data reveal a shoulder developing with increasing $L$ . [72.1.5.4] We have fitted the right tail using this theoretical prediction. [72.1.5.5] The fit is shown as a guide to the eye in Figure 6 using eq. (23) with appropriate fit parameters. [72.1.5.6] Because of the shift parameter $C$ in eq. (20) the predicted plateau at 16 appears for fully MCS-converged simulations at larger values of $L$ .

Figure 7: Tail analysis for $T=1.5$ . The solid lines represent the standard normal (Gaussian) distribution given in eq. (24).

Figure 8: Rescaled order parameter distributions $p_{0}(x)$ for $T=1.5$ and $L=16,32,64$ for fixed and periodic boundary conditions. (Only right half is shown for periodic boundary conditions).

[72.1.6.1] Figure 7 for $T=1.5$ shows some important results. [72.2.0.1] First consider the left tails in the upper row. Near the peak (i.e. for small $x$ ), the curves approach Gaussian behaviour as one would expect from the central limit theorem [13]. [72.2.0.2] However, there is only a relatively narrow regime over which Gaussian behaviour is seen. [72.2.0.3] In the intermediate range of $x$ the curves for the left tail show a plateau occuring at $0.5$ corresponding to a fat stretched exponential tail $\sim\exp(-\sqrt{x})$ . [72.2.0.4] In Figure 8 we show the full rescaled order parameter distributions at $T=1.5$ . [72.2.0.5] In the upper right hand corner of Figure 8 one sees a narrow Gaussian peak near $x=0$ . [72.2.0.6] It is followed by a stretched exponential tail. [72.2.0.7] For periodic boundary conditions the stretched exponential tail crosses over into a flat bottom. [72.2.0.8] For fixed boundary conditions the same stretched exponential tail is cutoff by a cutoff function. [72.2.0.9] The stretched exponential tail represents the well known droplet regime found analytically by Shlosman [29]. [72.2.0.10] We found this stretched exponential tail in the distributions for the low temperatures all the way up to the critical temperature. [72.2.0.11] Finally in the far tail regime the cutoff function lets the curves diverge to infinity for fixed boundary conditions. [72.2.0.12] In the case of periodic boundary conditions the order parameter distribution becomes a small constant corresponding to the value zero in our way of plotting the data. [72.2.0.13] This is again a well known phenomenon [29], reflecting phase coexistence on finite lattices governed by strip-like spin configurations.

[72.2.1.1] Next we turn to the right tails at $T=1.5$ . [72.2.1.2] Because the temperature is very low the magnetization is close to unity. [72.2.1.3] Therefore a peak develops only for larger system sizes explaining the absence of data for $L=16$ . [72.2.1.4] Even for $L=32$ only five data points exist to the right of the peak, and hence we can only conclude that the right tails seem to approach Gaussian behaviour with increasing $L$ .

D Convergence estimate

[72.2.2.1] Our results above show clearly that system sizes up to $L=64$ do not allow to determine the order parameter distribution at criticality. [72.2.2.2] Even away from criticality such system sizes are not sufficient to estimate the true Gaussian behaviour of the tails that has to emergein an infinite system.

[72.2.3.1] It is therefore of interest to estimate the values of $L$ that would suffice to obtain the true order parameter distribution at the critical point. We discuss two ad hoc methods for such an estimate.

[72.2.4.1] In the first method we extrapolate the peaks in Figures 2 and 3, and demand that $\Delta$ be smaller than some small threshold, e.g. $\Delta<0.1$ . [72.2.4.2] In Figures 9 and 10 we show extrapolations of $\Delta$ based on power law and logarithmic fits. [72.2.4.3] We were unable to fit the data to an exponential fit function.

Figure 9: Fit and extrapolation using power-law fit function $\Delta=aL^{b}$ for fixed (triangles) and periodic (circles) boundary conditions. The fit parameters are $a=1.039(35)$ and $b=-0.0891(98)$ for fixed boundary conditions and $a=0.993(47)$ and $b=-0.118(15)$ for periodic boundary conditions. The errors in brackets represent 95% confidence intervals. They are represented in the figure as dotted lines following the fit (solid line).

Figure 10: Fit and extrapolation using a logarithmic fit function $\Delta=a\log(1/L)+b$ for fixed (triangles) and periodic (circles) boundary conditions. The fit parameters are $a=0.0694(34)$ , $b=1.006(12)$ for fixed boundary conditions and $a=0.077(15)$ , $b=0.930(55)$ for periodic boundary conditions. The errors in brackets represent 95% confidence intervals. They are represented in the figure as dotted lines following the fit (solid line).

[72.2.5.1] We emphasize once more that the data points for $L=128$ are not MCS-converged and hence not fully reliable [72.2.5.2] . We also emphasize that the extrapolations are not meant to be accurate. [72.2.5.3] Their only purpose is to provide [page 73, §0] an order of magnitude estimate for the values of $L$ that we believe are needed to find the true $L$ -converged critical order parameter distribution. [73.1.0.1] Extrapolations from both, periodic and fixed boundary conditions, give values larger than $L\approx 10^{5}$ . [73.1.0.2] Of course these values increase further if one demands that the threshold value for $\Delta$ is smaller than $0.1$ .

[73.1.1.1] A second method to estimate which values of $L$ are needed for the true $L$ -converged form of the critical order parameter distribution is to demand that the critical data collapse should extend up to values of around 10 in the scaling variable $x=mL^{{1/8}}$ . [73.1.1.2] The range over which the data collapse determines the range over which the critical scaling function can be considered to be known. [73.1.1.3] From Figure 6 one sees that this range is only of the order $x\approx 1$ in our simulations. [73.1.1.4] A range of $x\approx 10$ might still be too small if one wants to decide whether or not the distribution develops algebraic tails. [73.1.1.5] Using a value of $x\approx 10$ as a lower bound and remembering that $|m|\leq 1$ one finds $L\approx 10^{8}$ . [73.2.0.1] Again this value is very high and in qualitative agreement with the high values found from the first extrapolation method.

[73.2.1.1] Although we have analysed only periodic and fixed boundary conditions here, some other boundary condition could be more relevant for such a study. [73.2.1.2] At criticality, a fully converged distribution must correspond to a value of $\Delta(T_{c},L)$ close to zero. [73.2.1.3] We find $\Delta(T_{c},64)$ for periodic boundary condition to be smaller than $\Delta(T_{c},64)$ for fixed boundary condition. [73.2.1.4] So, perhaps, some other boundary condition may give the true critical order parameter distribution at a lower value of $L$ than the extrapolated system sizes anticipated for these boundary conditions from the above analysis.

Authors:	R. Hilfer, B. Biswal, H.G. Mattutis, W. Janke
originally published in:	Physical Review E68, 046123 (2003)
originally submitted:	January 28th, 2003