Variational Loop Vertex Expansion for Cumulants

Abstract: Extending recent advances on constructive quantum field theory, we study cumulants of one of the simplest matrix models in the regime of bounded rank. We analyze both ordinary cumulants and scalar cumulants, which arise from the Weingarten calculus and are essential to the topological expansion in quantum field theory. Our results are valid for arbitrarily large positive coupling and provide new instances where techniques such as the variational approach are applied to cu- mulants.

• The paper extends LVE from free energies to cumulants in quartic matrix models, providing explicit convergent expansions and uniform analytic bounds.
• It introduces a variational approach using LVE trees with cilia and Weingarten calculus to accurately capture connected correlations.
• The method guarantees uniform Borel summability and a controlled 1/N topological expansion, offering new insights into non-perturbative regimes.

Variational Loop Vertex Expansion for Cumulants: An Expert Synthesis

Background and Motivation

The Loop Vertex Expansion (LVE) has established itself as a robust constructive method for addressing non-Gaussian random matrix and field models, notably providing convergent expansions well-suited to rigorous control over large $N$ limits and analytic structure. The current work extends the LVE’s reach by systematically analyzing cumulants—not merely the free energy—in quartic matrix models of bounded rank. This extension is critical since cumulants, which encode connected correlations (Schwinger functions), are central observables in both classical and quantum random matrix models, with deep implications for topological expansions and universality.

Model Definition and Analytic Domains

The work focuses on a single-matrix model with quartic interaction, parameterized as:

${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$

Here, $M$ is a complex $N\times N$ matrix and $J$ ($J^\dagger$) are external sources. $\lambda$ is considered complex, extending analyticity considerations to domains characterized by cardioid and pacman-like regions in the complex plane for uniform $N$ control.

The study simultaneously treats "ordinary" and "scalar cumulants," the latter being key in the matrix Weingarten calculus and the topological expansion. Scalar cumulants are defined as coefficients in the expansion of cumulant observables, decomposed over pair partitions and permutation group structures, leading to trace-invariant observables with a precise topological $1/N$ scaling.

Theoretical Advances

LVE for Cumulants

Prior results established LVE for partition functions and free energies. This work pushes forward by executing a variational LVE for higher-order cumulants, proving analyticity and providing explicit uniform bounds. The cumulants are expressed as convergent expansions over LVE trees (decorated with "cilia" marking the sources), with amplitudes constructed via intermediate field representations and non-Gaussian Gaussian integration (including explicit source dependence).

Main Structural Results

Let $\mathfrak{K}_{\{abcd\}}^{\mathcal{K}}(\lambda,N)$ denote the cumulant of order ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$0. The principal expansion is:

${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$1

where the sum runs over LVE trees with ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$2 cilia. Each amplitude is represented as a finite-dimensional integral and a sum over discrete combinatorial objects (set partitions and permutations).

Additional structure is revealed by expressing scalar cumulants using Weingarten functions reflecting the invariant integration over ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$3, encoding nontrivial topology in cycles of permutations.

Analyticity, Bounds, and Resummation

A strong nonperturbative result is established: the cumulants and scalar cumulants are analytic for ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$4 in a cardioid (or, via variational extension, for ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$5 in an even larger domain ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$6), uniformly in ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$7. This control is crucial for both physical applications and rigorous large ${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$8 expansions. Explicit factorized bounds are given for every term:

${\cal Z}[\lambda,N;J,J^+] = \int dM\, \exp\left\{ \sqrt{N}\left[\Tr(J M^\dagger) + \Tr(M J^\dagger)\right] - \Tr(M M^\dagger) - \frac{\lambda}{2N} \Tr(M M^\dagger M M^\dagger) \right\}$9

Uniform Borel summability follows for all cumulants, with rescaled cumulants being Borel-LeRoy summable uniformly in $M$0. This realizes the Nevanlinna-Sokal paradigm for matrix cumulants in the presence of arbitrary sources.

The topological expansion is also controlled, with the dominant terms indexed by surface genus and number of broken faces. The paper gives precise bounds on the remainder in the topological $M$1 expansion and for the perturbative expansion up to order $M$2.

Combinatorial and Technical Lemmas

A series of lemmas, many adapting and extending those from Sazonov [Saz], rigorously establish bounds on corner operators, combinatorics of LVE tree amplitudes, as well as control over contribution multiplicities. This includes explicit counting formulas for trees with fixed cilia, bounds on leaves/internal vertices, and control over the sums in both "dense" and "sparse" tree regimes, ensuring analyticity up to large orders.

Implications and Theoretical Impact

The results lead to several key theoretical implications:

• Constructive control in non-perturbative domains: The variational LVE grants constructive access to matrix cumulants in strong coupling regimes ($M$3 finite and large), far from mere perturbative expansions.
• Uniform Borel summability: The uniform bounds guarantee resummability and analytic continuation, extending the classical Borel summability of the free energy to all cumulants.
• Topological expansion with rigorous control: The method provides explicit terms and uniform bounds for the $M$4 expansion, with precise combinatorial weights related to the genus expansion and ribbon graphs induced by cumulant partitioning.
• Weingarten calculus and universality: The scalar cumulant expressions clarify the link to Weingarten functions on unitary groups, underpinning the combinatorics of matrix model observables essential for random geometry and quantum chaos applications.

Future Directions

Natural directions for further research include:

• Extension to multi-matrix and tensor models: While this work directly addresses the single-matrix (or fixed rank) quartic model, the method sets the stage for treating more general interactions and fields, including tensor models with higher-order invariants [RivTenTr3].
• Applications to constructive QFT and random geometry: The uniform analyticity and explicit bounds enable rigorous treatments of dual geometries (ribbon and ciliated graphs), relevant for quantum gravity and random surface theory.
• Algorithmic and numerical realizations: The explicit representation of cumulants as sums over LVE trees and Weingarten calculus facilitates systematic numerical explorations, including non-perturbative regimes where traditional Monte Carlo or combinatorial enumeration struggle.
• Analysis of universality and scaling limits: Understanding universality classes for both free energy and higher-order statistics under large $M$5 or double scaling may be informed by the tight analytic and combinatorial control established here.

Conclusion

This work rigorously develops a variational Loop Vertex Expansion for cumulants in quartic matrix models with sources, synthesizing and extending approaches from both constructive field theory and combinatorial matrix analysis. Strong analyticity, explicit uniform bounds, Borel summability, and precise topological expansions are established for all cumulants. These results deepen constructive control over matrix field theory observables, support further generalizations to broader models, and enable a unified framework connecting field-theoretic, combinatorial, and algebraic perspectives on non-Gaussian matrix integrals.