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Asymptotic expansion of the multi-orientable random tensor model

Eric Fusy, Adrian Tanasa

TL;DR

This work analyzes the large-$N$ expansion of the three-dimensional multi-orientable (MO) tensor model by extending Gurău–Schaeffer scheme techniques to MO graphs. It defines melon-free cores and reduced schemes, proving finiteness of schemes at fixed degree $\delta$ and identifying dominant schemes associated with rooted binary trees; it then derives generating functions and precise asymptotics for the number of rooted MO-graphs of degree $\delta$, showing planarity becomes overwhelmingly likely in the dominant sector. The results elucidate the combinatorial structure of MO graphs, expose the role of broken chain-vertices in determining singular behavior, and provide a concrete path toward a double scaling limit and higher-dimensional generalizations. These findings have potential impact on understanding universality and phase structure in tensor models and on connecting tensor combinatorics with analytic combinatorics methods.

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 expansion 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.

Asymptotic expansion of the multi-orientable random tensor model

TL;DR

This work analyzes the large- expansion of the three-dimensional multi-orientable (MO) tensor model by extending Gurău–Schaeffer scheme techniques to MO graphs. It defines melon-free cores and reduced schemes, proving finiteness of schemes at fixed degree and identifying dominant schemes associated with rooted binary trees; it then derives generating functions and precise asymptotics for the number of rooted MO-graphs of degree , showing planarity becomes overwhelmingly likely in the dominant sector. The results elucidate the combinatorial structure of MO graphs, expose the role of broken chain-vertices in determining singular behavior, and provide a concrete path toward a double scaling limit and higher-dimensional generalizations. These findings have potential impact on understanding universality and phase structure in tensor models and on connecting tensor combinatorics with analytic combinatorics methods.

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 expansion 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.

Paper Structure

This paper contains 17 sections, 19 theorems, 25 equations, 24 figures.

Table of Contents

  1. Introduction and motivation
  2. Preliminaries
  3. Multi-orientable tensor graphs
  4. The degree of an MO-graph
  5. MO-graphs as oriented 4-regular maps.
  6. Regular colored graphs for $D=3$ as a subfamily of MO-graphs
  7. From MO-graphs to schemes
  8. Extracting the melon-free core of a rooted MO-graph
  9. Extracting the scheme of a rooted melon-free MO-graph
  10. Finiteness of the set of reduced schemes at fixed degree
  11. Analysis of the removal of dipoles and of chain-vertices
  12. Proof of Lemma \ref{['lem:bound1']}
  13. Proof of Lemma \ref{['lem:bound2']}
  14. MO-graphs of degree $1/2$
  15. Dominant schemes
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $G$ be a connected MO-graph. Let $V$ be the number of vertices, $F_s$ the number of straight faces, and for each $p\geq 1$, $F_s^{(p)}$ the number of straight faces of length $2p$ in $G$. Then the quantity $\Lambda:=\sum_{p\geq 1}(p-2)\cdot F_s^{(p)}+2=V-2F_s+2$ satisfies where $\delta$ and $g$ are respectively the degree and the genus of $G$. Hence $\Lambda\leq 2\delta$.

Figures (24)

  • Figure 1: An MO-graph, with the 3 types of strands (left strands in blue, straight strands in green, right strands in red).
  • Figure 2: The three jackets of the MO graph in Figure \ref{['fig:exemple_MO_graph']}.
  • Figure 3: The MO-graph shown in Figure \ref{['fig:exemple_MO_graph']}, represented as an oriented 4-regular map.
  • Figure 4: Left: a rooted colored graph in $3$ dimensions. Right: the induced rooted MO-graph.
  • Figure 5: Left: a melon. Right: a triple edge that does not form a melon.
  • ...and 19 more figures

Theorems & Definitions (46)

  • Claim 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Claim 3.1
  • ...and 36 more