Corrections to Classical Matrix Ensemble Moments, Non-Crossing Annular Pairings, and Ribbon Graphs

TL;DR

The paper develops a topological-combinatorial framework linking matrix-ensemble moment corrections to non-crossing annular pairings and ribbon graphs. It establishes explicit bijections between ribbon-graph classes on surfaces (real projective plane, torus, Klein bottle) and non-crossing annular pairings that encode 1/N and 1/N^2 corrections for GOE/GUE moments, and extends these ideas to Laguerre ensembles via bipartite ribbon graphs and annular permutations. The main contributions include precise bijections: b_1(n) with NC_2^δ(n,-n), a_1(n) with NC_2^T(n), and b_2(n) with NC_2^K(n), plus analogous Laguerre results with tilde variants. This establishes a unifying combinatorial-topological picture for low-order infinitesimal fluctuations in classical random matrix ensembles, and provides a pathway to higher-order infinitesimal freeness in free probability contexts.

Abstract

We elucidate a bijection between ribbon graphs on the real projective plane and non-crossing annular pairings that relate to the $1/N$ correction term of the GOE and LOE spectral moments. We also derive analogous objects for the $1/N^2$ correction terms of said moments and their equivalents for the GUE and LUE.

Figures (14)

• Figure 1: These ribbon graphs encode how one must identify summation indices when computing $\mathbb E\mathop{\mathrm{Tr}}\nolimits H^4$ for the GUE. Their weights are respectively $N^3$, $N^3$, and $N$.
• Figure 2: These ribbon graphs contribute to $\mathbb E\mathop{\mathrm{Tr}}\nolimits H^4$ for the GOE. They are respectively of Euler genus $1$, $1$, and $2$.
• Figure 3: These ribbon graphs contribute to $\mathbb E\mathop{\mathrm{Tr}}\nolimits W^2$ for the LUE. Assigning the colour black (white) to vertices labelled by $i_1,i_2$ ($j_1,j_2$) and letting $M=cN$, we see that these graphs are respectively weighted $NM^2=c^2N^3$ and $N^2M=cN^3$.
• Figure 4: This is a hypermap on $A=[5]$ with vertex permutation $\gamma=(1,2,3)(4,5)$ and hyperedge permutation $\pi=(1,2,4)(3,5)$. Equivalently, it is a bipartite map with $|A|=n=5$ edges where the cycles of $\gamma$ represent the black vertices and the cycles of $\pi$ represent white vertices. Note that $\pi$ is non-crossing with respect to $\gamma$ because this hypermap is connected and represents a cell decomposition of the genus zero sphere with faces given by the cycles of $\pi^{-1}\gamma=(1)(2,5)(3,4)$.
• Figure 5: The non-crossing disk and annular pairings with respect to $\gamma=(1,\dots,6)$ (left) and $\gamma=(1,2,3,4)(5,6,7,8)$ (middle) of Example \ref{['Example:non crossing pairings']} with the symmetric non-crossing annular pairing $\pi=(1,-4)(4,-1)(2,3)(-2,-3)\in\mathop{\mathrm{NC}}\nolimits_2^\delta(4,-4)$ (right). The cycles $(1,8)$, $(4,5)$, $(1,-4)$, and $(4,-1)$ connecting the inner and outer circles of the annuli are called through strings.

Theorems & Definitions (37)

• definition 1: GOE, GUE, LOE, and LUE
• remark 1
• theorem 1: Isserlis--Wick
• proposition 1: GUE moments
• remark 2
• proposition 2: GOE moments
• proposition 3: LUE moments
• proposition 4: LOE moments
• definition 2: Partitions and pairings
• definition 3: Join