Testing the Gaussian expansion method in exactly solvable matrix models

TL;DR

This paper evaluates the Gaussian expansion method using exactly solvable matrix models, clarifying its properties, limitations, and applications in understanding nonperturbative phenomena and space-time dimensionality.

Contribution

It applies the Gaussian expansion to solvable matrix models, introduces a consistent prescription for including linear terms, and analyzes multiple solutions related to space-time dimensionality.

Findings

The method works with various potentials, including unbounded and double-well types.

Multiple large-N solutions can be identified, corresponding to different physical phases.

The approach clarifies issues in dynamical space-time generation in matrix models.

Abstract

The Gaussian expansion has been developed since early 80s as a powerful analytical method, which enables nonperturbative studies of various systems using `perturbative' calculations. Recently the method has been used to suggest that 4d space-time is generated dynamically in a matrix model formulation of superstring theory. Here we clarify the nature of the method by applying it to exactly solvable one-matrix models with various kinds of potential including the ones unbounded from below and of the double-well type. We also formulate a prescription to include a linear term in the Gaussian action in a way consistent with the loop expansion, and test it in some concrete examples. We discuss a case where we obtain two distinct plateaus in the parameter space of the Gaussian action, corresponding to different large-N solutions. This clarifies the situation encountered in the dynamical determination of the space-time dimensionality in the previous works.