Asymptotic expansion for transport maps between laws of multimatrix models

Abstract: We study the large-$N$ behavior of random matrix tuples $Y^N = (Y_1^N,\dots,Y_d^N)$ with joint density proportional to $e^{-N^2 V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1/N^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S^{N}t){t \geq 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction.

• The paper introduces an explicit asymptotic expansion for expectations of trace functions in multimatrix random matrix models.
• It develops a novel Banach-space framework for non-commutative smooth functions, enabling analytic control of transport maps and precise error bounds.
• The results extend strong operator-norm convergence from GUE to broader convex potentials, significantly advancing universality in random matrix theory.

Asymptotic Expansion of Transport Maps in Multimatrix Models

Introduction and Background

This work develops a comprehensive theory of asymptotic expansions for transport maps between probability laws associated to multimatrix random matrix models. Specifically, consider random matrix $d$-tuples $Y^N = (Y_1^N, \ldots, Y_d^N)$ of size $N$ with measures proportional to $\exp(-N^2 V)$, where $V$ is a convex, non-commutative function of the variables. Such models generalize classical Gibbs measures, with non-commutative "potentials" appearing in combinatorial map enumeration and topological applications.

The authors focus on two main aspects: establishing an explicit asymptotic expansion (in powers of $1/N^2$) for expectations of traces of non-commutative smooth functions and studying the expansions for transport maps that push forward the law of a tuple of independent GUE matrices to the law of $Y^N$. Central to their approach is a general framework of non-commutative smooth function spaces compatible with the analytic structure required for both the expansion and the transport construction.

Non-Commutative Smooth Functions and Asymptotic Expansion

A central technical innovation is the development of Banach-space projective tensor-norm based function spaces $\mathcal{C}^k$ for non-commutative smooth functions. These generalize operator-valued polynomial and smooth functional calculus, capturing trace polynomials and multi-variable higher-order expansions. The introduced differential calculus, with free difference quotients and cyclic gradients, is tailored for systematic control over Schwinger–Dyson equations and higher-order cumulant expansions. This technical machinery is strictly necessary for analytic control beyond the combinatorial genus expansion context.

The main technical result is an explicit expansion

$\mathbb{E} \tr_N \big[ f(Y^N) \big] = \tau(f_0(x)) + \frac{1}{N^2}\tau(f_1(x)) + \cdots + \frac{1}{N^{2k}}\tau(f_k(x)) + \mathcal{O}(N^{-2k-2}),$

where $f$ is a non-commutative smooth function and the terms $Y^N = (Y_1^N, \ldots, Y_d^N)$0 are determined by universal (but complex) differential operations on $Y^N = (Y_1^N, \ldots, Y_d^N)$1 and the potential $Y^N = (Y_1^N, \ldots, Y_d^N)$2 evaluated in a free semicircular system $Y^N = (Y_1^N, \ldots, Y_d^N)$3 ("free Gibbs law"). The explicit error bounds are detailed in Theorem 1 of the paper. These expansions strictly generalize previous results that assumed polynomial or analytic potentials, now allowing far greater functional generality and regularity control.

Construction of Transport Maps

Building on analytic methods, the authors link the finite-$Y^N = (Y_1^N, \ldots, Y_d^N)$4 and infinite-dimensional ('free') settings via nonlinear transport maps $Y^N = (Y_1^N, \ldots, Y_d^N)$5. These maps interpolate between GUE laws and those induced via non-quadratic, possibly non-analytic, convex potentials, using a time-dependent path $Y^N = (Y_1^N, \ldots, Y_d^N)$6 connecting the quadratic and final potential. The transport $Y^N = (Y_1^N, \ldots, Y_d^N)$7 is constructed via an infinitesimal equation involving the gradient of the pseudo-inverse of the generator of the associated Langevin diffusion: $Y^N = (Y_1^N, \ldots, Y_d^N)$8 where $Y^N = (Y_1^N, \ldots, Y_d^N)$9 involves the heat semigroup generated by the non-commutative Laplacian. The asymptotic expansion of $N$0 is controlled using the smooth function framework, with explicit higher-order error bounds on operator norm balls (Theorem 2).

A notable contribution is the analytic derivation of these transport expansions, as opposed to previous arguments relying on more algebraic or combinatorial structures. This analytic approach is crucial for subsequent extensions (norm convergence, function composition, conditional expectations).

Strong Convergence and Operator-Norm Results

A critical application of the asymptotic expansion for transport is the extension of strong convergence (in the operator norm) results to general multimatrix models with convex and sufficiently regular potentials. Previous results were limited to special ensembles and combinatorial methods; here, convergence of the operator norms of smooth functions in $N$1 to those in the "free" limit is established: $N$2 where $N$3 is the $N$4 limit of the transport maps (Theorem 3). This result generalizes the classical strong convergence theory for GUE and related ensembles to a far wider class of potentials, leveraging the transport expansion rather than explicit resolvent or Schwinger–Dyson methods.

Technical Innovations and Analytic Tools

Key advances include:

• Construction and closure of the $N$5 function spaces under composition, differentiation, conditional expectation, and evaluation at free or random matrix inputs.
• Sharp control of higher-order derivatives in non-commutative settings, using Banach projective tensor products.
• Quantitative analytic (non-combinatorial) estimation of all error terms, crucial for the convergence and norm topology results.
• Systematic use of the Langevin SDE via a smooth functional calculus on matrix-valued Brownian motions.
• Transfer of large-$N$6 operator-norm convergence from GUE to general ensembles via explicit, controlled transport.

Implications and Prospects

This framework, by providing analytic control over asymptotic expansions for a wide variety of non-commutative functions and convex potentials, significantly advances the analytic theory of random matrix models and free probability. Practically, it enables rigorous passage from Gaussian to non-Gaussian matrix models in strong topology, extending universality results and norm convergence in the context of operator algebras and free Gibbs states.

Theoretically, the analytic techniques and function space methods developed here set a foundation for the possible future study of non-convex or even less regular potentials, perturbative expansions near non-Gaussian points, and models relevant for quantum information and mathematical physics. The stability under composition and conditioning is particularly promising for applications to random matrix integrals and the structure of large random operator algebras.

Future developments may address extending these expansions to the inclusion of tensor products with deterministic matrices and analyzing non-linear or non-elliptic generalizations of the Langevin-type SDE, as well as further connections to optimal transport in non-commutative settings.

Conclusion

This work establishes a robust analytic framework for asymptotic expansions of traces and transport maps in multimatrix random matrix models, based on a sophisticated calculus of non-commutative smooth functions. The results yield new operator-norm convergence theorems, provide analytic alternatives to combinatorial genus expansion, and equip the field with tools to approach highly regular multimatrix models well beyond the reach of previous combinatorial and algebraic methods.