Here are some applications of free probability of random matrices:

- *Neural networks:* The asymptotic freeness assumption plays a fundamental role in the study of the propagation of spectral distributions through the layers of a deep neural network, see <A HREF="https://arxiv.org/abs/2103.13466">Asymptotic Freeness of Layerwise Jacobians Caused by Invariance of Multilayer Perceptron</A> (2021) and <A HREF="https://proceedings.neurips.cc/paper_files/paper/2022/file/10826a1a80f816ea98d559d7c7a97973-Paper-Conference.pdf">Free Probability for predicting the performance of feed-forward fully connected neural networks</A> (2022).
- *Wireless communication:* The asymptotic capacity limit of a multi-antenna channel can be calculated from the an asymptotic free probability assumptions, see <A HREF="https://arxiv.org/abs/1007.2119">Free Probability based Capacity Calculation of Multiantenna Gaussian Fading Channels with Cochannel Interference</A> (2010).
- *Quantum information:* Free probability is helpful to understand the problem of the quantificatio of the degree of entanglement of a multi-qubit system, see <A HREF="https://www.semanticscholar.org/paper/APPLICATIONS-OF-FREE-PROBABILITY-TO-QUANTUM-THEORY-Fukuda/37669ee1621fc6ea25437f253466c06a97ce7123">Applications of free probability to quantum information theory</A> (2014).
- *Statistical physics:* The free energy of exclusion processes can be calculated using tools from free probability, see <A HREF="https://arxiv.org/abs/2211.01710">Bernoulli variables, classical exclusion processes and free probability</A> (2022).