Permutation Invariant Gaussian Matrix Models
This work provides a theoretical foundation for permutation invariant matrix models in computational linguistics, but it is incremental as it extends a previously solved 5-parameter case to a more general 13-parameter one.
The paper solves the general 13-parameter Gaussian matrix model using representation theory, enabling the identification of constraints for a convergent measure and the computation of expectation values for polynomial invariants up to quartic order.
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.