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

TL;DR

This paper establishes a connection between finite-dimensional matrix control problems and their infinite-dimensional free-probability counterparts, providing insights into large deviations and non-commutative control theory.

Contribution

It proves that the non-commutative value function describes the large-$n$ limit of matrix control problems under convexity assumptions.

Findings

Non-commutative value function captures large-$n$ limit.

Provides a new perspective on the Laplace principle.

Connects finite matrix control with free probability.

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.