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

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.

16 pages RevTeX, 14 figures; v2: Algorithm for the Weingarten model corrected; v3: published version

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Cited by in corpus (4)

• Incorporating Dynamic Mean-Field Theory into Diagrammatic Monte Carlo
• A method for resummation of perturbative series based on the stochastic solution of Schwinger-Dyson equations
• Diagrammatic Monte-Carlo for weak-coupling expansion of non-Abelian lattice field theories: large-N U(N)xU(N) principal chiral model
• Metropolis updates for Diagrammatic Monte-Carlo algorithms from Schwinger-Dyson equations