×

Permutation invariant Gaussian matrix models. (English) Zbl 1430.81065

Summary: 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.

MSC:

81T32 Matrix models and tensor models for quantum field theory
20M35 Semigroups in automata theory, linguistics, etc.
20B30 Symmetric groups
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
15B52 Random matrices (algebraic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Harris, Z., Mathematical Structures of Language (1968), Wiley · Zbl 0195.02202
[2] Firth, J. R., A synopsis of linguistic theory 1930-1955, (Studies in Linguistic Analysis (1957))
[3] Coecke, B.; Sadrzadeh, M.; Clark, S., Mathematical foundations for a compositional distributional model of meaning, Lambek Festschrift. Lambek Festschrift, Linguist. Anal., 36, 345-384 (2010)
[4] Grefenstette, E.; Sadrzadeh, M., Concrete models and empirical evaluations for a categorical compositional distributional model of meaning, Comput. Linguist., 41, 71-118 (2015)
[5] Maillard, J.; Clark, S.; Grefenstette, E., A type-driven tensor-based semantics for CCG, (Proceedings of the Type Theory and Natural Language Semantics Workshop (2014), EACL)
[6] Baroni, M.; Bernardi, R.; Zamparelli, R., Frege in space: a program of compositional distributional semantics, Linguist. Issues Lang. Technol., 9 (2014)
[7] Kartsaklis, D.; Sadrzadeh, M.; Pulman, S., A unified sentence space for categorical distributional-compositional semantics: Theory and experiments, (Proceedings of 24th International Conference on Computational Linguistics (COLING): Posters. Proceedings of 24th International Conference on Computational Linguistics (COLING): Posters, Mumbai, India (2012)), 549-558
[8] Kartsaklis, D.; Ramgoolam, S.; Sadrzadeh, M., Linguistic matrix theory, Ann. Inst. Henri Poincaré D (2019) · Zbl 1447.91125
[9] Kartsaklis, D.; Ramgoolam, S.; Sadrzadeh, M., Linguistic matrix theory, Video at · Zbl 1447.91125
[11] Hamermesh, M., Group Theory and Its Application to Physical Problems (1962), Dover · Zbl 0151.34101
[12] Fulton, W.; Harris, J., Representation Theory: A First Course (2004), Springer
[13] Zee, A., Group Theory in a Nutshell for Physicists (2016), Princeton University Press · Zbl 1346.20001
[14] Naimark, M. A.; Stern, A. I., Theory of Group Representations, A Series of Comprehensive Studies in Mathematics, vol. 246 (1982) · Zbl 0484.22018
[15] de Mello Koch, R.; Ramgoolam, S., Free field primaries in general dimensions: counting and construction with rings and modules, J. High Energy Phys., 1808, Article 088 pp. (2018) · Zbl 1396.81186
[16] Online notes by David Zhang
[18] He, Y. H.; Jejjala, V.; Nelson, B. D., hep-th
[19] Benkart, Georgia; Halverson, Tom, Partition algebras and the invariant theory of the symmetric group · Zbl 1421.05094
[20] Gabriel, F., Combinatorial theory of permutation-invariant random matrices I: partitions, geometry and renormalization
[21] Gabriel, F., Combinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processes
[22] Au, Benson; Cébron, Guillaume; Dahlqvist, Antoine; Gabriel, Franck; Male, Camille, Large permutation invariant random matrices are asymptotically free over the diagonal · Zbl 1480.15040
[23] Male, Camille, Traffic distributions and independence: permutation invariant random matrices and the three notions of independence · Zbl 1440.15037
[24] Peskin, M.; Schroder, D. V., An Introduction to Quantum Field Theory (1995), Taylor and Francis Group
[25] Zee, A., Quantum Field Theory in a Nutshell (2010), Princeton University Press · Zbl 1277.81001
[27] ’t Hooft, G., A planar diagram theory for strong interactions, Nucl. Phys. B, 72, 461 (1974)
[28] Maldacena, J. M., The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.. Int. J. Theor. Phys., Adv. Theor. Math. Phys., 2, 231 (1998) · Zbl 0914.53047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.