Random matrix theory has played a major role in several areas of pure and applied mathematics, as well as statistics, physics, and computer science. This lecture aims to describe the intrinsic freeness phenomenon and how it provides new easy-to-use sharp non-asymptotic bounds on the spectrum of general random matrices. We will also present a couple of illustrative applications in high dimensional statistical inference. This article accompanies a lecture that will be given by the author at the International Congress of Mathematicians in Philadelphia in the Summer of 2026.

翻译：随机矩阵理论在纯数学与应用数学的多个领域，以及统计学、物理学和计算机科学中均发挥了重要作用。本讲座旨在阐述内蕴自由性现象，并说明它如何为一般随机矩阵的谱提供新颖、易用且尖锐的非渐近界。我们还将展示该理论在高维统计推断中的若干示例性应用。本文为作者将于2026年夏季在费城国际数学家大会上所作讲座的配套文稿。