Title: Linguistic Matrix Theory,
Abstract: Recent research in computational linguistics has developed algorithms which
associate matrices with adjectives and verbs, based on the distribution of
words in a corpus of text. These matrices are linear operators on a vector
space of context words. They are used to construct the meaning of composite
expressions from that of the elementary constituents, forming part of a
compositional distributional approach to semantics. We propose a Matrix Theory
approach to this data, based on permutation symmetry along with Gaussian
weights and their perturbations. A simple Gaussian model is tested against word
matrices created from a large corpus of text. We characterize the cubic and
quartic departures from the model, which we propose, alongside the Gaussian
parameters, as signatures for comparison of linguistic corpora. We propose that
perturbed Gaussian models with permutation symmetry provide a promising
framework for characterizing the nature of universality in the statistical
properties of word matrices. The matrix theory framework developed here
exploits the view of statistics as zero dimensional perturbative quantum field
theory. It perceives language as a physical system realizing a universality
class of matrix statistics characterized by permutation symmetry.