Large $n$-limit of matrix control problems and non-commutative controls

TL;DR

The paper establishes that the large-n limit of finite-dimensional matrix stochastic control problems is governed by a non-commutative, free-probability-based value function defined on tracial von Neumann algebras. By combining a rigorous discretization approach, $E$-convexity assumptions, and Voiculescu’s asymptotic freeness, it proves convergence of finite-n value functions to the infinite-dimensional non-commutative limit. The framework unifies mean-field-type control, random matrix theory, and free probability, and yields a Laplace principle for convex functionals in large deviations for random matrices. This provides a principled way to analyze large systems with non-commutative randomness and non-local interactions in optimization problems, with potential implications for quantum control and large-scale stochastic systems.

Abstract

Building on the free-probability stochastic control framework introduced in arXiv:2502.17329, we connect optimal control problems for $n \times n$ random matrix ensembles with their infinite-dimensional, free-probability analogues. Under natural convexity hypotheses, we prove that the non-commutative value function captures the large-$n$ limit of the corresponding finite-matrix control problems. As an application, we give a new perspective on the Laplace principle for convex functionals in the theory of large deviations for random matrices.

Theorems & Definitions (42)

• Theorem 2.1
• Lemma 3.1: Lemma 3.9 in 2025viscosity
• Lemma 3.2: Lemma 3.10 in 2025viscosity
• Lemma 3.3
• proof
• Lemma 3.4
• Lemma 3.5
• Definition 3.6
• Theorem 3.7
• Remark 4.1