Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph

The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the $q$-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model ($q=2$) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for $q=2$ contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation ($q=1$) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for $q=1$ instead of $q=2$. This demonstrates the percolation effects in the FK Ising model on the complete graph.

Figure 1

The probability distribution of ${\mathcal{C}}_1$, the size of the largest cluster. Here $f_{X_1}\left(x\right)$ (top) and $f_{X_1^{\prime }}\left(x\right)$ are respectively the probability density functions of $X_1={\mathcal{C}}_1/V^{3/4}$ and $X_1^{\prime }={\mathcal{C}}_1/V^{2/3}$. The black dashed line in the top figure shows the rigorous limiting distribution $f_{X_1}^{\infty }\left(x\right)$, shown in Eq. (3).

Figure 2

The probability distribution of ${\mathcal{C}}_2^{\mathrm{P}}$, the size of the second largest cluster in the percolation sector. Here $f_{X_2^{\mathrm{P}}}\left(x\right)$ is the probability density function of $X_2^{\mathrm{P}}={\mathcal{C}}_2^{\mathrm{P}}/V^{2/3}$. The inset plots the average $C_2^{\mathrm{P}}$ versus $V$.

Figure 3

The cluster-size distribution $n_{\mathrm{P}}(s,V)$ in the percolation sector. The slope of the black dashed line is $-5/2$. The inset plots in log-log scale $n_{\mathrm{P}}(s,V)s^{5/2}$ versus $s/V^{2/3}$. It shows that the scaling function $\tilde{n}\left(x\right)$ is consistent with $1/\sqrt{2\pi }$ if $x\ll 1$.

Figure 4

The probability distribution of ${\mathcal{C}}_2^{\mathrm{I}}$, the size of the second-largest cluster in the Ising sector. Here $f_{X_2^{\mathrm{I}}}\left(x\right)$ is the probability density function of $X_2^{\mathrm{I}}:={\mathcal{C}}_2^{\mathrm{I}}/(\sqrt{V}logV)$. The inset plots the average $C_2^{\mathrm{I}}/(\sqrt{V}logV)$ versus $V$.

Figure 5

Log-log plot of the cluster-size distribution $n_{\mathrm{I}}(s,V)$ versus $s$ in the Ising sector. The slope of the black dashed line is $-5/2$. The inset shows the log-log plot of $n_{\mathrm{I}}(s,V)s^{5/2}$ versus $s/(\sqrt{V}logV)$, which implies that the scaling function is consistent with $1/\sqrt{2\pi }$ when $s/(\sqrt{V}logV)\ll 1$.

Figure 6

Plot to show the leading correction term of the size of the largest cluster $C_1$. A straight line with slope $1/12$ is to guide the eye.

Figure 7

Plot of the distribution of ${\mathcal{C}}_2$, size of the second-largest cluster. Here $f_{X_2}\left(x\right)$ (top) and $f_{X_2^{\prime }}\left(x\right)$ (bottom) are respectively the probability density function of $X_2={\mathcal{C}}_2/\left(\sqrt{V}{log}_{10}V\right)$ and $X_2^{\prime }={\mathcal{C}}_2/V^{2/3}$. The curve in the top figure plots $h\left(x\right)$, shown in Eq. (A1), which is the limiting distribution of $X_2$ presented in Ref. [5].

Figure 8

Log-log plot of $n(s,V)$ versus $s$ for various systems. The slopes of the two dashed lines are $5/2$ and $7/3$, respectively, corresponding the Fisher exponent taking the percolation and Ising values. The inset plot $n(s,V)s^{5/2}$ versus $s/V^{3/4}$ on a log-log scale, and the data suggest the scaling function is constant at $1/\sqrt{2\pi }$ when $s\ll V^{3/4}$.

Figure 9

The $(K_{\mathrm{P}},K)$ diagram and its associated RG flow. The line $K_{\mathrm{P}}=K$ corresponds to the FK Ising model, and $(K_{\mathrm{P},\mathrm{c}},K_{\mathrm{c}})$ is the critical point.

Figure 10

Plot to show the distribution of ${\mathcal{C}}_2^{\mathrm{I}}$, the size of the second largest cluster in the Ising sector. Here $X_2^{\mathrm{I}}:={\mathcal{C}}_2^{\mathrm{I}}/\left(\sqrt{V}{log}_{10}V\right)$ and $f_{X_2^{\mathrm{I}}}\left(x\right)$ is its probability density function. It can be seen that, for $x<x_0$ with $x_0\approx 0.2$, data from various systems are observed to collapse onto a straight line with slope $-1/36$.

Figure 11

Finite-size scaling of the reduced susceptibility ${\chi }^{\prime }$ at the critical point $p_c$.

Figure 12

The finite-size scaling of the reduced susceptibility ${\chi }^{\prime }$ in (a) low-$T$ and (b) high-$T$ critical windows, and the thermodynamic-limit behavior of ${\chi }^{\prime }$ in (c) low-$T$ and (d) high-$T$ regions. The same range of vertical axis applies to panels (a) and (b) and to panels (c) and (d).