Related papers: Lecture Notes on "Free Probability Theory"
Notes for a Course on Probability and Statistics: L1: Elements of Probability; L2: Bayesian Inference; L3: Monte Carlo Methods
We investigate the implications of free probability for random matrices. From rules for calculating all possible joint moments of two free random matrices, we develop a notion of partial freeness which is quantified by the breakdown of…
These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.
Free probability provides a framework for describing correlations between non-commuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework…
These notes are based on my four lectures given at the Newton Institute in April 2004 during the Recent Perspectives in Random Matrix Theory and Number Theory Workshop. Their purpose is to introduce the reader to the analytic number theory…
These lecture notes have been developed for the course Computational Social Choice of the Artificial Intelligence MSc programme at the University of Groningen. They cover mathematical and algorithmic aspects of voting theory.
These lecture notes are based on lectures given in 2019 Saint-Flour Probability School.
This work gives an overview of analytic tools for the design, analysis, and modelling of communication systems which can be described by linear vector channels such as y = Hx+z where the number of components in each vector is large. Tools…
A four part series of lectures on the connection of statistical mechanics and quantum field theory. The general principles relating statistical mechanics and the path integral formulation of quantum field theory are presented in the first…
We solve two longstanding major problems in Free Probability. This is achieved by generalising the theory to one with values in arbitrary commutative algebras. We prove the existence of the multi-variable $S$-transform, and show that it is…
This article, which is substantially motivated by the previous joint work with J. McKay [8], establishes the analytic analogues of the relations we found free probability has with Witt vectors. Therefore, we first present a novel analytic…
We present a formal operator-theoretic framework for analyzing Transformer-based language models using free probability theory. By modeling token embeddings and attention mechanisms as self-adjoint operators in a tracial \( W^*…
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…
One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges…
Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the…
The concept of freeness was introduced by Voiculescu in the context of operator algebras. Later it was observed that it is also relevant for large random matrices. We will show how the combination of various free probability results with a…
Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free,…
Utilizing the framework of free probability, we analyze the spectral and operator statistics of the Rosenzweig-Porter random matrix ensembles, which exhibit a rich phase structure encompassing ergodic, fractal, and localized regimes.…
Lecture notes for the Yale Computer Science course CPSC 4690/5690 Randomized Algorithms. Suitable for use as a supplementary text for an introductory graduate or advanced undergraduate course on randomized algorithms. Discusses tools from…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…