Abstract
Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expanion in $N$, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.
Highlights
Introduction and motivationRandom tensor models generalize in dimension three the celebrated matrix models
In this article we have shown that, as in the colored model [9], the combinatorial study of MO-graphs can be done by extraction of so-called schemes, such that there are finitely many schemes in each fixed degree δ
We have identified the dominant schemes in each degree, whose shapes are naturally associated to rooted binary trees
Summary
Random tensor models (see [11] for a recent review) generalize in dimension three (and higher) the celebrated matrix models (see, for example, [3] for a review). A interesting combinatorial approach for the study of these colored graphs was proposed recently by Gurau and Schaeffer in [9], where they analyze in detail the structure of colored graphs of fixed degree and perform exact and asymptotic enumeration This analysis relies on the reduction of colored graphs to some terminal forms, called schemes. Let us mention that the analysis of this paper may further allow for the implementation of the the so-called double scaling limit for the MO tensor model This is a important mechanism for matrix models (see again [3]), making it possible to take, in a correlated way, the double limit N → ∞ and z → zc where z is a variable counting the number of vertices of the graph and zc is some critical point of the generating function of schemes of a given degree
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