Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regular colored graphs of positive degree

Razvan Gurau, Gilles Schaeffer

TL;DR

This work develops a structural theory for regular colored graphs of fixed positive degree, showing that for any fixed dimension $D\ge3$ and degree $\delta$, the set of reduced schemes is finite and the generating function is algebraic with a positive radius of convergence. By decomposing graphs into melon-free cores and attached melonic subgraphs and introducing chains and chain-vertices, the authors derive an exact enumeration formula $H^0_{\delta}(z)=T(z)\sum_{\tilde{S}\in\tilde{\cal S}^0_{\delta}} G_{\tilde{S}}(zT(z)^{D+1})$, with $T(z)$ defined by $T(z)=1+zT(z)^{D+1}$. They identify dominant schemes that maximize a linear objective ${\bf B}$ and analyze their singular behavior near the critical point $z_0=D^D/(D+1)^{D+1}$, which leads to a double scaling limit in colored tensor models; this limit is summable for $3\le D\le5$ but not for $D\ge6$, revealing regime-dependent continuum behavior. The results connect combinatorial structure with probabilistic scaling and provide a roadmap for understanding continuum limits of random colored triangulations in tensor models. The work also highlights open questions about exponential bounds on reduced schemes and the geometry of the resulting continuum spaces. Overall, the paper delivers a rigorous framework that unifies exact enumeration, asymptotic analysis, and scaling limits for colored graphs of fixed degree.

Abstract

Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of regular colored graphs of fixed positive degree and perform their exact and asymptotic enumeration. In particular we show that the generating function of the family of graphs of fixed degree is an algebraic series with a positive radius of convergence, independant of the degree. We describe the singular behavior of this series near its dominant singularity, and use the results to establish the double scaling limit of colored tensor models.

Regular colored graphs of positive degree

TL;DR

This work develops a structural theory for regular colored graphs of fixed positive degree, showing that for any fixed dimension and degree , the set of reduced schemes is finite and the generating function is algebraic with a positive radius of convergence. By decomposing graphs into melon-free cores and attached melonic subgraphs and introducing chains and chain-vertices, the authors derive an exact enumeration formula , with defined by . They identify dominant schemes that maximize a linear objective and analyze their singular behavior near the critical point , which leads to a double scaling limit in colored tensor models; this limit is summable for but not for , revealing regime-dependent continuum behavior. The results connect combinatorial structure with probabilistic scaling and provide a roadmap for understanding continuum limits of random colored triangulations in tensor models. The work also highlights open questions about exponential bounds on reduced schemes and the geometry of the resulting continuum spaces. Overall, the paper delivers a rigorous framework that unifies exact enumeration, asymptotic analysis, and scaling limits for colored graphs of fixed degree.

Abstract

Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of regular colored graphs of fixed positive degree and perform their exact and asymptotic enumeration. In particular we show that the generating function of the family of graphs of fixed degree is an algebraic series with a positive radius of convergence, independant of the degree. We describe the singular behavior of this series near its dominant singularity, and use the results to establish the double scaling limit of colored tensor models.

Paper Structure

This paper contains 41 sections, 39 theorems, 61 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Context.
  3. Our results.
  4. Discussion.
  5. Organization of the paper
  6. Notation and generalities on colored graphs
  7. The core of a rooted colored graph
  8. First attempts to define the core
  9. 2-edge-cuts in colored graphs.
  10. Melons and melonic graphs
  11. The core decomposition
  12. Chains
  13. Maximal chains
  14. Classification of chains
  15. Schemes
  16. ...and 26 more sections

Key Result

theorem thmcountertheorem

For any fixed dimension $D\geq3$ and degree $\delta\geq 0$, there exist a finite set $\tilde{\cal S}^0_{\delta}$ of reduced schemes of degree $\delta$ and root edge of color $0$, and triples $( P_{\tilde{S}}(u), {\bf U}_{\tilde{S}}, {\bf B}_{\tilde{S}} )_{ \tilde{S}\in \tilde{\cal S}^0_{\delta} }$ c where $T(z)$ is the unique power series solution of the equation:

Figures (23)

  • Figure 1: A colored graph $G$ (where the root edge is represented as crossed), the open colored graph $\mathrm{op}(G)$, and a face $(0,3)$ of $G$.
  • Figure 2: A ring graph $R^c$ and a trivial open graph $\mathbb{E}^c=\textrm{op}(R^c)$.
  • Figure 3: Examples of open colored subgraphs of an open colored graph.
  • Figure 4: The removal of a melon
  • Figure 5: A 2-edge-cut, the set-cut of the root edge and the corresponding open and closed components.
  • ...and 18 more figures

Theorems & Definitions (81)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • proposition thmcounterproposition: color
  • proof
  • corollary thmcountercorollary
  • proof
  • ...and 71 more