Finite $N$ precursors of the free cumulants

TL;DR

The paper constructs finite-$N$ precursors $K_n^{(N)}(A)$ and generalized precursors $K_oldsymboleta^{(N)}(A)$ from the HCIZ integral, providing a finite-$N$ analogue of free cumulants with an additive property under averaging over sums of $ ext{U}(N)$ conjugacy orbits. It proves that as $N oty$, these precursors converge to the standard free cumulants $ ext{κ}_n$ (and $ ext{κ}_{oldsymboleta}$ in the product form), while at finite $N$ they exhibit a rich topological expansion controlled by monotone Hurwitz numbers, yielding explicit $1/N^2$ corrections and an invertible moment–cumulant relation. The authors also develop a generating function for the precursors, establish a Wick-type averaging for Gaussian ensembles, and introduce an orbit coproduct that encodes how invariant polynomials decompose under the addition of conjugacy orbits. Extending to averaged functionals of random matrices, they connect with Collins–Gurau–Lionni cumulants and demonstrate additivity under convolution, linking algebraic and probabilistic pictures. The framework unifies several strands—from Weingarten calculus to Hurwitz counts—and provides practical tools for finite-$N$ matrix models and finite free probability, with implications for Horn-type spectral problems and beyond.

Abstract

We study $\mathrm{U}(N)$ invariant polynomials on the space of $N\times N$ matrices first introduced by Capitaine and Casalis, that are precursors of free cumulants in various respects. First, they are polynomials of deterministic matrices, that are not yet evaluated over some probability law, contrary to what is usually meant by cumulants. Secondly, they converge towards the algebraic expression of free cumulants in terms of moments as $N\to \infty$, with $1/N^2$ corrections expressed in terms of monotone Hurwitz numbers. Their most crucial property is their additivity with respect to averaging over sums of $\mathrm{U}(N)$ conjugacy orbits, providing a finite $N$ version of the well-known additivity of free cumulants in free probability. Finally, they extend several properties of free cumulants at finite $N$, including a Wick rule for their average over a Gaussian weight and their appearance in various matrix integrals. Building on the additivity property of these precursors, we also define and compute a coproduct describing the behaviour of general invariant polynomials with respect to the addition of $\mathrm{U}(N)$ conjugacy orbits, as well as their expectation values on sums of $\mathrm{U}(N)$-invariant random matrices. In our construction, a central role is played by the so-called HCIZ integral, both for the definition of the precursors and for the derivation of their properties.

Figures (1)

• Figure 1: A numerical experiment with $4\times 4$ matrices $A$ and $B$ of regularly spaced spectrum on $[-1,1]$ and a sample of 200,000 Haar-distributed matrices $U$, showing the histograms of $\delta \kappa_4(U)$ (left), $\delta K_4(U)$ (middle) and $\delta f_4(U)$ (right). The improvement of the latter two with respect to the former is manifest: smaller skewness, distribution closer to Gaussian, etc. The broader range of values of $\delta K_4$ with respect to $\delta \kappa_4$ is due to the prefactor $N^4(N^2 + 1) /((N^2 - 1) (N^2 - 4) (N^2 - 9)$ in \ref{['Kmrel']}.

Theorems & Definitions (42)

• Definition 1
• Remark 1
• Theorem 1
• Definition 2
• Theorem 2
• Remark 2
• Proposition 1
• Proposition 2
• Remark 3
• proof