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Combinatorial aspects of Parraud's asymptotic expansion for GUE matrices

David Jekel

TL;DR

The paper proves a new combinatorial derivation of Parraud's $1/N^2$-expansion for moments of polynomials in independent GUE matrices, connecting the analytic derivative framework to the genus expansion. It introduces crossing derivatives based on free difference quotients and integrates over auxiliary parameters to obtain a compact, iterated expansion that mirrors Taylor-type remainders. The approach provides a transparent link between map-based genus enumeration and Parraud's derivative formula, clarifying how derivative operations correspond to crossings in maps. The work also outlines pathways to extend the combinatorial proof to smooth non-polynomial functions and to other ensembles, highlighting broader applications in random matrix theory and free probability.

Abstract

We give a new combinatorial proof of Parraud's formula for the asymptotic expansion in powers of $1/N^2$ for the expected trace of polynomials of several independent $N \times N$ GUE matrices, which expresses the result using a mixture of free difference quotients, introducing new freely independent semicircular variables, and integration with respect to parameters. Our approach streamlines the statement of the formula while clarifying its relationship to the combinatorial genus expansion.

Combinatorial aspects of Parraud's asymptotic expansion for GUE matrices

TL;DR

The paper proves a new combinatorial derivation of Parraud's -expansion for moments of polynomials in independent GUE matrices, connecting the analytic derivative framework to the genus expansion. It introduces crossing derivatives based on free difference quotients and integrates over auxiliary parameters to obtain a compact, iterated expansion that mirrors Taylor-type remainders. The approach provides a transparent link between map-based genus enumeration and Parraud's derivative formula, clarifying how derivative operations correspond to crossings in maps. The work also outlines pathways to extend the combinatorial proof to smooth non-polynomial functions and to other ensembles, highlighting broader applications in random matrix theory and free probability.

Abstract

We give a new combinatorial proof of Parraud's formula for the asymptotic expansion in powers of for the expected trace of polynomials of several independent GUE matrices, which expresses the result using a mixture of free difference quotients, introducing new freely independent semicircular variables, and integration with respect to parameters. Our approach streamlines the statement of the formula while clarifying its relationship to the combinatorial genus expansion.
Paper Structure (15 sections, 10 theorems, 107 equations, 4 figures)

This paper contains 15 sections, 10 theorems, 107 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of Parraud's formula
  4. Iterated Parraud's formula
  5. Preliminaries
  6. GUE random matrices
  7. Free semicircular families
  8. Genus expansion for mixtures of GUE and semicirculars
  9. Permutations
  10. Statement of the genus expansion
  11. Operations on permutations
  12. Inductive proof of the genus expansion
  13. Combinatorial proof of Parraud's formula
  14. Interpolation between GUE and semicircular
  15. Combinatorial evaluation of the crossing derivative

Key Result

Theorem 1.1

With the notation above, we have where $T_{V,W}^{\mathop{\mathrm{cr}}\nolimits}$ is the linear operator $\mathbb{M}_N\langle V \oplus W\rangle \to \mathbb{M}_N\langle V \oplus [(V \oplus W) \otimes \mathbb{R}^6]\rangle$ described by Definition def: T cross below. Here the tensor products of the indexing vector spaces are taken over

Figures (4)

  • Figure 3.1: Circles (solid) constructed from a permutation $\sigma = (12345)(678)$ and connecting curves (dotted) constructed from a permutation $\pi = (17)(24)(35)(68)$. Here $(24)$ and $(35)$ are a $\sigma$-cross and $(17)$ is a $\sigma$-bridge.
  • Figure 3.2: Local picture of the construction of a surface as in Remark \ref{['rem: surface genus']}. The solid edges come from the permutation $\sigma$ while the dotted edges come from the permutation $\pi$. On each corner, we glue disks of types (1), (2), (3) as described in the remark.
  • Figure 3.3: A cycle $\sigma$ and a transposition $\tau$ on a set $S$ produce a permutation $\sigma \lefthalfcap \tau$ consisting of two cycles. We label the two pieces (a) and (b).
  • Figure 3.4: A permutation $\sigma$ consisting of two cycles and a transposition $\tau$ connecting them produce a permutation $\sigma \lefthalfcap \tau$ consisting of one cycle.

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.4
  • Remark 1.5: Real versus complex tensor products
  • Definition 1.9
  • Definition 1.10
  • Example 1.11: Crossing derivative of a sixth-degree monomial
  • Definition 1.12
  • Corollary 1.13
  • Definition 2.1
  • Definition 2.4
  • ...and 31 more