Gibbs sampling of complex-valued distributions

A new technique is explored for the Monte Carlo sampling of complex-valued distributions. The method is based on a heat bath approach where the conditional probability is replaced by a positive representation of it on the complex plane. Efficient ways to construct such representations are also introduced. The performance of the algorithm is tested on small and large lattices with a $\lambda {\varphi }^4$ theory with quadratic nearest-neighbor complex coupling. The method works for moderate complex couplings, reproducing reweighting and complex Langevin results and fulfilling various Schwinger-Dyson relations.

Figure 1

One-branch paths on the complex plane $\varphi$, parametrized by $0\le u\le 1$, for the periodic complex probability with action $S(\varphi )=a{e}^{i\varphi }+b{e}^{-i\varphi }$, with $a=0.5$ and $b=0.15i$. From bottom to top, the paths start at $\varphi =0.2i,0.3i,0.5i,i,2i$. The three middle paths are periodic and so any of them provides a one-branch representation of the complex probability.

Figure 2

Left: Function $Q_1(x)=Q_2(x)$ for $P(x)=\delta (x)+{\delta }^{\prime }(x)$, using the optimal value $Y=1.58$. Right: Functions $Q_1(x)$ (solid line) and $Q_2(x)=Q_1(-x)$ (dashed line) for the two-branch representation of the action $S(x)=\beta x^2/2$ with $\beta =1+i$ using the optimal value $Y=0.7$.

Figure 3

Functions $C(x,Y)$ and $S(x,Y)$ for $Y=0.5$ (larger amplitude), 1 and 2 (smaller amplitude).

Figure 4

Functions $Q_1(x)$ (solid line) and $Q_2(x)$ (dashed line) for the two-branch representation of the periodic action $S(x)=\beta \cos (x)+imx$. Left: $\beta =1-i$ and $m=1$, with $Y=1.17$. Right: $\beta =i$, $m=0$, with $Y=0.93$.

Figure 5

Representation of $P\propto g(x_1)g(x_2)g(x_1-x_2)$ for $g(x)=1+2a\cos (x)$ with $a=i$, using $Y_2=3Y_1$. The function $Q_1(x_1,x_2)$ is displayed on $[-\pi ,\pi ]\times [-\pi ,\pi ]$ for $Y_2=4.43$. This is the optimal value, that is, $\min Q_1=0$.

Figure 6

Same as Fig. 5 with $a=1$ and $Y_2=3Y_1=2.17$.

Figure 7

Monte Carlo estimates of $O_1$, $O_2$, $I_1$ and $I_2$ for a $3^3$ lattice with $\beta =0.25i$, from RW (green squares), CL (red rhombuses) and CHB (blue triangles). Each point represents one of the 20 independent runs. To guide the eye ellipses are extracted from mean values and variance matrices of each cloud. They are scaled to enclose 68% of the Gaussian probability.

Figure 8

Metastability of the RW calculation. Monte Carlo estimates of $O_1$, $O_2$, $I_1$ and $I_2$ for an $8^4$ lattice with $\beta =0.5i$, from RW (green squares), CL (red rhombuses) and CHB (blue triangles). For this lattice and coupling the RW estimate displays a net deviation (as well as enhanced dispersion). The incorrect RW estimates displayed actually reproduce the expectation values of the auxiliary action $\mathrm{Re}S[\varphi ]$ (see text). Correct RW results would follow from using a sufficiently large number of sweeps.

Figure 9

Influence of the parameter $Y_s$. The points represent estimates of $I_2$ from 20 runs with 68% ellipses, for an $8^3$ lattice with $\beta =0.5i$: CHB with $Y_s=2$ (green squares), CHB with $Y_s=5$ (black disks), CHB with $Y_s=\infty$ (blue triangles), and CL (red rhombuses).

Figure 10

Typical runs for $L=20$ and $L=40$ with $\beta =0.5i$ in an $8^3$ lattice. The $10^5$ sweeps are arranged in 100 batches of 1000 sweeps each. The noisy thin solid (blue) line represents $\mathrm{Re}⟨I_1⟩$ for each batch. The thick solid (black) line represents the average over all batches to the right (and so computed at later simulation times). The thick dashed (red) line shows the effect of removing batches which are off by $4\sigma$ ($L=20$) or $8\sigma$ ($L=40$). The high peak in “$L=40$” reaches 0.66.

Figure 11

Monte Carlo estimates of $O_1$, $O_2$, $I_1$, and $I_2$ for an $8^3$ lattice with $\beta =0.5i$, using $L=40$ and $K=12$. CL is represented by red rhombuses, and CHB by black disks. The blue triangles are from CHB upon removal of batches (subsets of 1000 sweeps) lying beyond $8\sigma$ for $\mathrm{Re}⟨I_1⟩$ (i.e., ${\mathrm{CHB}}^{**}$ in Table 2). As can be seen, the impact of the removal is small on $O_1$ and $O_2$, while $I_1$ and $I_2$ are brought closer to zero.