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Large N Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension d ≥ 2

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dc.contributor.author Bonzom, V.
dc.date.accessioned 2019-02-16T09:10:47Z
dc.date.available 2019-02-16T09:10:47Z
dc.date.issued 2016
dc.identifier.citation Large N Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension d ≥ 2 / V. Bonzom // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 49 назв. — англ. uk_UA
dc.identifier.issn 1815-0659
dc.identifier.other 2010 Mathematics Subject Classification: 05C10; 05C75; 83C45; 81T18; 83C27
dc.identifier.other DOI:10.3842/SIGMA.2016.073
dc.identifier.uri http://dspace.nbuv.gov.ua/handle/123456789/147839
dc.description.abstract We review an approach which aims at studying discrete (pseudo-)manifolds in dimension d≥2 and called random tensor models. More specifically, we insist on generalizing the two-dimensional notion of p-angulations to higher dimensions. To do so, we consider families of triangulations built out of simplices with colored faces. Those simplices can be glued to form new building blocks, called bubbles which are pseudo-manifolds with boundaries. Bubbles can in turn be glued together to form triangulations. The main challenge is to classify the triangulations built from a given set of bubbles with respect to their numbers of bubbles and simplices of codimension two. While the colored triangulations which maximize the number of simplices of codimension two at fixed number of simplices are series-parallel objects called melonic triangulations, this is not always true anymore when restricting attention to colored triangulations built from specific bubbles. This opens up the possibility of new universality classes of colored triangulations. We present three existing strategies to find those universality classes. The first two strategies consist in building new bubbles from old ones for which the problem can be solved. The third strategy is a bijection between those colored triangulations and stuffed, edge-colored maps, which are some sort of hypermaps whose hyperedges are replaced with edge-colored maps. We then show that the present approach can lead to enumeration results and identification of universality classes, by working out the example of quartic tensor models. They feature a tree-like phase, a planar phase similar to two-dimensional quantum gravity and a phase transition between them which is interpreted as a proliferation of baby universes. While this work is written in the context of random tensors, it is almost exclusively of combinatorial nature and we hope it is accessible to interested readers who are not familiar with random matrices, tensors and quantum field theory. uk_UA
dc.description.sponsorship This paper is a contribution to the Special Issue on Tensor Models, Formalism and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/Tensor Models.html. This research was supported by the ANR MetACOnc project ANR-15-CE40-0014. uk_UA
dc.language.iso en uk_UA
dc.publisher Інститут математики НАН України uk_UA
dc.relation.ispartof Symmetry, Integrability and Geometry: Methods and Applications
dc.title Large N Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension d ≥ 2 uk_UA
dc.type Article uk_UA
dc.status published earlier uk_UA

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