Markov chain sampling of the O(n) loop models on the infinite plane

A numerical method was recently proposed in Herdeiro and Doyon [Phys. Rev. E 94, 043322 (2016)] showing a precise sampling of the infinite plane two-dimensional critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the O(n) loop gas models for $n\in (1,2]$. We argue that even though the Gibbs measure is nonlocal, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the O(n) models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.

Figure 1

We are here looking at a corner of the square domain $A$, for which boundaries $\partial A$ are given by the thick straight black lines. The olive and khaki areas are two ordered domains with the opposite sign thus the white lines between are arcs of loops entering the partition function. This graph illustrates a case study for a (Wolff) flip of the $\mathcal{C}$ virtual cluster from olive to khaki. $ⓐ$ and $ⓑ$ represent two different, incompatible, connection sets of information beyond the boundary. In the text, we discuss how each set of information changes the acceptance probability of flipping $\mathcal{C}$.

Figure 2

The scaling weights of the energy and spin operators have similar profiles (both increase with $n$); this means that correlations decay faster with increasing $n$. Regarding the spin operator, a faster decay with larger $n$ makes sense: The larger $n$ the more energetically favorable it is to create a new loop and thus induce a flip in the value of ${\sigma }_{\mathbf{i}}{\sigma }_{\mathbf{i}+\mathbf{x}}$ and thus a decrease of its statistical average.

Figure 3

Graphs of $\langle {\sigma }_{\mathbf{i}}^L{\sigma }_{\mathbf{j}}^L\rangle \left(n\right)$ for $n=1.25,1.5,1.75,2$ and for separations $|\mathbf{i}-\mathbf{j}|$ in $[1,120]$. The power law behavior is manifest. The exponents and offsets fitted are presented in Eqs. (12) and (11). The fits were done by ${\chi }^2$ minimization on a $x\to a{x}^b$ template function.

Figure 4

Graphs of $\left(\langle {\varepsilon }_{\mathbf{i}}^L{\varepsilon }_{\mathbf{j}}^L\rangle -{\langle {\varepsilon }^L\rangle }^2\right)\left(n\right)$ for $n=1.25,1.5,1.75,2$ and for separations $|\mathbf{i}-\mathbf{j}|$ in $[5,20]$. For simplicity, all insertions are made along the horizontal direction only. We observe a satisfying power law profile. Only the $n=2$ curve shows some noise in the tail. Fits were performed to extract exponent, amplitude, and disconnected parts (12) and (11). These fits were done by ${\chi }^2$ minimization on a function $x\to a{x}^b+c$.

Figure 5

Plots of $F_{\sigma \sigma \sigma \sigma }(\eta =\overline{\eta },\kappa \left(n\right))$ for $n=1.25,1.5,1.75$, and 2. The respective ${\chi }^2$ values are 0.355, 0.579, 0.671, and 0.614. They share a $p$ value of 1.

Figure 6

Picture of $A$ being dilated to $\lambda A$, $\lambda \approx 1.2$ here. $\partial \left({\lambda }^{-1}A\right)$, the dashed contour square, gets mapped to $\partial A$, the continuous contour square. Before integrating out $\left(\partial A,\partial \left(\lambda A\right)\right)$—the light gray area—by cropping it, all information in the loops crossing $\partial A$ needs to be stored in a new connection map object. Cases $①$ and $②$ are described in the text.

Figure 7

The red line is the graph of the theoretical expression given by (7). The orange points are the fitted scaling exponent values. Each point is the exponent result of a power law fit in the lattice linear size. The uncertainty bars are included but thinner than the data points' width. Here as well we fitted by ${\chi }^2$ minimization. The agreement is obvious and extremely accurate.

Figure 8

On the $x$ axis, the separation between the two spin insertions. On the $y$ axis, the value of the average spin correlator. Both axes are log scaled. The power law behavior seems obvious, the fitted exponent is $2{\mathrm{\Delta }}_{\sigma }\left(1.5\right)=0.44180\left(5\right)$ to be compared to the theoretical expectation of $0.43892\cdots$ The fit ${\chi }^2$ is $\approx {10}^{-6}$, a very low value.

Figure 9

Monitoring the mixing of a $O(n=2)$ MCMC. Along the $x$ axis, the chronology of the evolution steps. The vertical dashed lines show the occurrences of a lattice dilation. The $y$ axis is duplicated. The blue curve shows the value of the fitted ${\mathrm{\Delta }}_{\sigma }\left(t\right)$ with axis markers on the left side, whereas the red curve shows the fits ${\chi }^2$ with units on the right; the latter is log scaled. The two horizontal lines show the value of ${\mathrm{\Delta }}_{\sigma }$ for the Ising and the $O\left(2\right)$ model, orange and chocolate, respectively.