Permutation Invariant Gaussian Matrix Models

TL;DR

This paper extends permutation invariant Gaussian matrix models by solving a more general 13-parameter case using representation theory, enabling detailed computation of matrix invariants and ensuring measure convergence.

Contribution

It introduces a representation theoretic solution to the 13-parameter Gaussian matrix model, generalizing previous work and providing explicit formulas for invariants.

Findings

Solved the 13-parameter Gaussian matrix model using representation theory

Expressed matrix invariants in terms of model parameters

Ensured convergence and well-defined expectations for polynomial functions

Abstract

Permutation invariant Gaussian matrix models were recently developed for applications in computational linguistics. A 5-parameter family of models was solved. In this paper, we use a representation theoretic approach to solve the general 13-parameter Gaussian model, which can be viewed as a zero-dimensional quantum field theory. We express the two linear and eleven quadratic terms in the action in terms of representation theoretic parameters. These parameters are coefficients of simple quadratic expressions in terms of appropriate linear combinations of the matrix variables transforming in specific irreducible representations of the symmetric group $S_D$ where $D$ is the size of the matrices. They allow the identification of constraints which ensure a convergent Gaussian measure and well-defined expectation values for polynomial functions of the random matrix at all orders. A graph-theoretic interpretation is known to allow the enumeration of permutation invariants of matrices at linear, quadratic and higher orders. We express the expectation values of all the quadratic graph-basis invariants and a selection of cubic and quartic invariants in terms of the representation theoretic parameters of the model.