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In around 1990 Voiculescu showed asymptotic freeness of certain random matrices, i.e., free independence when the matrix size goes to infinity. Since then this link between free probability and random matrices has been the subject of much research. For example, asymptotic freeness has been shown for various classes of random matrices.

Random matrices have various applications, though only some of them involve matrices increasing in size. I think it would be interesting and useful to have a list of applications where asymptotic freeness of random matrices is used. This to motivate the study of asymptotic freeness.

Wireless communication appears to be one such example: in their book "Random Matrix Theory and Wireless Communications", Tulino and Verdu cover random matrices, free probability, and applications to wireless communications. It seems it also finds application to neural networks, judging from a search on Google Scholar, though I wouldn't know how.

Are there any other, perhaps surprising, applications of asymptotic freeness of random matrices? Links to sources would be much appreciated.

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    $\begingroup$ Perhaps it should be noted that the Kreweras-Voiculescu partition polynomials of OEIS A134264, giving the free moments in terms of the free cumulants and the inverse set of partition polynomials of A350499, giving the free cumulants in terms of the free moments of free probability theory, play important roles in enumerative combinatorics related to the Weyl-Coxeter groups $A_n$ and analysis and group theory related to compositional inversion of power and Laurent series and, consequently, the Legendre transform, so important in quantum and statistical physics. $\endgroup$
    – Tom Copeland
    Commented Jan 21 at 16:47

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Here are some applications of free probability of random matrices:

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Here are a few additional references, in the same directions as in Carlo's answer.

Wireless communication:

Channel Estimation and Robust Detection for IQ Imbalanced Uplink Massive MIMO-OFDM With Adjustable Phase Shift Pilots (Chen, You, Lu, Gao and Xia ,2021)

Neural networks:

Demystifying Disagreement-on-the-Line in High Dimensions (Lee, Moniri, Huang, Dobriban and Hassan 2023)

The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization (Adlam and Pennington, 2020)

Approximate Message Passing algorithms for rotationally invariant matrices (Fan, 2022)

Statistical Physics:

General Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems (Pappalardi, Fritzsch and Prosen 2023)

Designs via Free Probability (Fava, Kurchan and Pappalardi, 2023)

Spectrum of subblocks of structured random matrices : A free probability approach (Bernard and Hruza, 2023)

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