Canonical partition functions for gauged permutation-invariant s-index tensor oscillators are expressed exactly as products of (1 - x^{LCM(...)})^{-...} factors, with a large-N critical Boltzmann factor x_c ~ log N/(s N^{s-1}) derived under a stated two-partition dominance conjecture.
Permutation invariant matrix statistics and computational language tasks
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abstract
The Linguistic Matrix Theory programme introduced by Kartsaklis, Ramgoolam and Sadrzadeh is an approach to the statistics of matrices that are generated in type-driven distributional semantics, based on permutation invariant polynomial functions which are regarded as the key observables encoding the significant statistics. In this paper we generalize the previous results on the approximate Gaussianity of matrix distributions arising from compositional distributional semantics. We also introduce a geometry of observable vectors for words, defined by exploiting the graph-theoretic basis for the permutation invariants and the statistical characteristics of the ensemble of matrices associated with the words. We describe successful applications of this unified framework to a number of tasks in computational linguistics, associated with the distinctions between synonyms, antonyms, hypernyms and hyponyms.
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Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle
Canonical partition functions for gauged permutation-invariant s-index tensor oscillators are expressed exactly as products of (1 - x^{LCM(...)})^{-...} factors, with a large-N critical Boltzmann factor x_c ~ log N/(s N^{s-1}) derived under a stated two-partition dominance conjecture.