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

TL;DR

This paper introduces a stochastic approach to solving Schwinger-Dyson equations in large-N quantum field theories using nonlinear random processes, enabling sampling of expectation values via stationary distributions, with applications to matrix models and sigma-models.

Contribution

The paper presents a novel stochastic method for solving Schwinger-Dyson equations in large-N theories, including techniques to handle divergences through self-consistent redefinitions.

Findings

Method successfully samples expectation values in matrix scalar theories.

Self-consistent redefinition handles divergences in weak-coupling regimes.

Illustrated on O(N) sigma-model and discussed for lattice gauge theories.

Abstract

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 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 the 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. 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 example of two-dimensional O(N) sigma-model. Extension to non-Abelian lattice gauge theories is discussed.