Permutation invariant Gaussian matrix models
Abstract
Permutation invariant Gaussian matrix models were recently developed for applications in computational linguistics. A 5parameter family of models was solved. In this paper, we use a representation theoretic approach to solve the general 13parameter Gaussian model, which can be viewed as a zerodimensional 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 welldefined expectation values for polynomial functions of the random matrix at all orders. A graphtheoretic 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 graphbasis invariants and a selection of cubic and quartic invariants in terms of the representation theoretic parameters of the model.
 Publication:

Nuclear Physics B
 Pub Date:
 August 2019
 DOI:
 10.1016/j.nuclphysb.2019.114682
 arXiv:
 arXiv:1809.07559
 Bibcode:
 2019NuPhB.94514682R
 Keywords:

 High Energy Physics  Theory;
 Computer Science  Computation and Language;
 Mathematical Physics;
 Mathematics  Representation Theory
 EPrint:
 47 pages. Revisionsmall changes in presentation to align with NPB published version, typos corrected