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Loop Vertex Representation for Cumulants, Part II: Weingarten Calculus

Vincent Rivasseau

TL;DR

This work develops a constructive, convergent expansion for scalar cumulants of even-order Weingarten-calculus-based random matrix ensembles within the loop vertex representation (LVR). By leveraging Fuss–Catalan combinatorics, contour-integral vertex methods, and Weingarten functions, the authors express cumulants as absolutely convergent sums over LVR trees with explicit bounds and analytic dependence on the coupling $\lambda$, plus controlled remainders. The main contributions include analyticity and Borel summability of the cumulant expansions for $1\le {\mathcal K} \le {\mathcal K}_{\max}$, with uniform-in-$N$ estimates and a rigorous link to the Borel–LeRoy–Nevanlinna–Sokal framework (via $q\to p-1$). This provides a robust, nonperturbative handle on large-$N$ limits for complex matrix models with higher-degree interactions, connecting loop-vertex techniques to 2D quantum gravity map combinatorics through Weingarten calculus.

Abstract

In this paper we construct scalar cumulants for stable random matrix models with single trace interactions of arbitrarily high even order by Weingarten calculus. We obtain explicit and convergent expansions for these scalar cumulants in the limit N tend to infinity.

Loop Vertex Representation for Cumulants, Part II: Weingarten Calculus

TL;DR

This work develops a constructive, convergent expansion for scalar cumulants of even-order Weingarten-calculus-based random matrix ensembles within the loop vertex representation (LVR). By leveraging Fuss–Catalan combinatorics, contour-integral vertex methods, and Weingarten functions, the authors express cumulants as absolutely convergent sums over LVR trees with explicit bounds and analytic dependence on the coupling , plus controlled remainders. The main contributions include analyticity and Borel summability of the cumulant expansions for , with uniform-in- estimates and a rigorous link to the Borel–LeRoy–Nevanlinna–Sokal framework (via ). This provides a robust, nonperturbative handle on large- limits for complex matrix models with higher-degree interactions, connecting loop-vertex techniques to 2D quantum gravity map combinatorics through Weingarten calculus.

Abstract

In this paper we construct scalar cumulants for stable random matrix models with single trace interactions of arbitrarily high even order by Weingarten calculus. We obtain explicit and convergent expansions for these scalar cumulants in the limit N tend to infinity.
Paper Structure (6 sections, 10 theorems, 45 equations, 10 figures)

This paper contains 6 sections, 10 theorems, 45 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Results
  4. Summary of KRS and KRS1
  5. Proof of Theorem \ref{['th2']}.
  6. Appendix: Borel-LeRoy-Nevanlinna-Sokal theorem

Key Result

Proposition 1

The scalar cumulants of order $2{\mathcal{K}}$ can be written as a sum over partitions of ${\mathcal{K}}$ and over two permutations of ${\mathcal{K}}$ elements: where ${\mathfrak{K}}^{\mathcal{K}} (\lambda,N)$ is defined by Definition cumulantsdef and where $\tau_{\pi}$ and $\xi_{\pi}$ are arbitrary permutations such that $\tau_{\pi}(\xi_{\pi})^{-1}$ has a cycle structure corresponding to the par

Figures (10)

  • Figure 1: In blue the cardioid domain considered in rivasseau2022cumulants, in red the cardioid domain considered in BGKL.
  • Figure 2: A LVR graph In the case $p=2$ with five vertices colored in black, four propagators colored in red, two loops colored in blue, two cilia and one broken face colored in gray.
  • Figure 3: A vertex with some of its corner operators (courtesy from KRS1). The label $k$ indicates the corresponding contour variable. The upper left corner between the two half-edges $\sqcup$ symbols contains three $(u-K)^{-1}$ operators with indices $k$, $k$ and $k+1$.
  • Figure 4: A tree of $n-1$ lines on $n$ loop vertices (depicted as rectangular boxes, hence here $n=5$) defines a forest of $n+1$ connected components or cycles ${\cal C}$ on the $2n$ elementary loops, since each vertex contains exactly two loops. To each such cycle corresponds a trace of a given product of operators in the LVR.
  • Figure 5: A Feynman graph of the $\mathop{\mathrm{Tr}}\nolimits (M^\dagger M )^3$ theory. All five vertices are 6-valent. One $M^\dagger$ field per vertex leads to a dotted line (for better visibility we showed as dotted the hooked point plus a large fraction of the propagator). We define in this case two connected components, namely two loop vertices.
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2: Weingarten definition
  • Proposition 1
  • Theorem 1
  • Definition 3
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proposition 3
  • ...and 3 more