Free probability theory

Free probability theory is a branch of mathematics that studies systems of random variables that are free from each other, similar to how independent variables behave in traditional probability. Developed by mathematician Daniil Kramkov and others, it provides tools to analyze noncommutative structures, such as those arising in quantum mechanics and random matrix theory. In essence, free probability helps us understand how these interconnected random elements interact, leading to new insights in fields like operator algebras and mathematical physics. It contrasts with classical probability by focusing on how randomness can coexist without influencing one another.