Schwinger-Dyson equations in large-$N$ quantum field theories and nonlinear random processes

We propose a stochastic method for solving Schwinger-Dyson equations in large-$N$ quantum field theories. Expectation values of single-trace operators are sampled by stationary probability distributions of the so-called nonlinear random processes. The set of all the histories of such processes corresponds to the set of all planar diagrams in the perturbative expansions of the expectation values of singlet operators. We illustrate the method on examples of the matrix-valued scalar field theory and the Weingarten model of random planar surfaces on the lattice. For theories with compact field variables, such as sigma models or non-Abelian lattice gauge theories, the method does not converge in the physically most interesting weak-coupling limit. In this case one can absorb the divergences into a self-consistent redefinition of expansion parameters. A stochastic solution of the self-consistency conditions can be implemented as a “memory” of the random process, so that some parameters of the process are estimated from its previous history. We illustrate this idea on the two-dimensional $O(N)$ sigma model. The extension to non-Abelian lattice gauge theories is discussed.

Figure 1

Schematic illustration of the structure of the Schwinger-Dyson equations (6). Dashed blobs denote the Green functions $G(k_1,\dots ,k_n)$ and empty blobs denote the free propagator $G_0(k)$.

Figure 2

Green functions $G_n(\lambda )$ in (18) versus the coupling constant $\lambda$ for $n=1,\dots ,5$, obtained after $N={10}^6$ discrete time steps of the algorithm described above at fixed $c=8$ and with $\mathcal{N}$ given by branch 1 of (20). The error bars are smaller than the symbols on the plot. Exact results of 6 are plotted with solid lines.

Figure 3

Autocorrelation time of the random process described in Sec. 4b as a function of the coupling constant $\lambda$ at fixed $c=8$ and for different choices of $\mathcal{N}$ in (20).

Figure 4

Mean stack size of the random process described in Sec. 4b as a function of the coupling constant $\lambda$ at fixed $c=8$ and for different choices of $\mathcal{N}$ in (20).

Figure 5

The quantity ${\xi }_n^s$ in (21) for $n=1$, 3, 5, as a function of the coupling constant $\lambda$ at fixed $c=8$ and for different choices of $\mathcal{N}$ in (20). “Br. 1,2” is for “Branch 1,2.”

Figure 6

Schematic illustration of the structure of the loop equations (24) in the Weingarten model (22). These equations should hold for any link (marked by a thick line) which belongs to the loop.

Figure 7

The observables $W_{1\times 0}\equiv W(\mu ,-\mu )$ ($1\times 0$ loop) and $W_{1\times 1}\equiv W(\mu ,\nu ,-\mu ,-\nu )$ ($1\times 1$ loop) in the Weingarten model as functions of the coupling constant $\beta$ for different dimensions $D$. The solid line corresponds to the first two terms in the perturbative expansion: $W(\mu ,-\mu )=1+2(D-1){\beta }^2+O({\beta }^4)$, $W(\mu ,\nu ,-\mu ,-\nu )=\beta +8(D-1){\beta }^3+O({\beta }^5)$. The data were obtained after ${10}^7$ iterations of the algorithm described above.

Figure 8

Two simple configurations of branched polymers on the lattice. The configurations on the left and on the right count as different configurations.

Figure 9

Mean rate of flattening events for the random process solving the loop equations in the Weingarten model as a function of the coupling constant $\beta$ at different dimensions $D$ and for different choices of $q$ in (27).

Figure 10

Mean stack size (upper panel) and mean length of the topmost loop in the stack (lower panel) for the random process solving the loop equations in the Weingarten model as a function of the coupling constant $\beta$ at different dimensions $D$ and for different choices of $q$ in (27).

Figure 11

The process of convergence of the random process with memory which solves the Schwinger-Dyson equations (30) for $D=2$. The quantity plotted is the estimate of the lattice spacing $a(\lambda ,t)$ after $t$ discrete time steps, with the exact result $a(\lambda ,t\to \infty )$ subtracted.

Figure 12

Numerical estimates of the lattice spacing as a function of the coupling constant for the two-dimensional large-$N$ $O(N)$ sigma model, compared with the exact result. The estimates were obtained using both algorithms A and B with different numbers of time steps.