Abstract: A quantum dynamical system can be described as an algebra of bounded operators in a non-commutative probability space equipped with a time evolution map. In free probability theory, quantum chaos is marked by “free independence” between observables at early and late times, causing certain statistical measures (cumulants) to vanish. Motivated by this, we study the statistics of a time-evolved operator in the Rosenzweig-Porter (RP) random matrix ensembles. The RP model displays a complex phase structure, namely the ergodic, fractal, and localized regimes, featuring two phase transitions. We show that the free probability approach is powerful enough to distinguish these three phases. The numerical results in the ergodic regime can be verified analytically using complex algebra tools such as the Cauchy transform and Stieltjes inversion methods. This talk is based on an ongoing work.

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