Three Lectures on Free Probability Jonathan Novak with Illustrations by Michael Lacroix
نویسنده
چکیده
0. Introduction 2 1. Lecture One: Discovering the Free World 2 1.1. Counting connected graphs 2 1.2. Cumulants and connectedness 4 1.3. Cumulants and independence 7 1.4. Central Limit Theorem by cumulants 9 1.5. Geometrically connected graphs 10 1.6. Non-crossing cumulants 12 1.7. Non-crossing independence 19 1.8. The medium is the message 21 1.9. A brief history of the free world 22 2. Lecture Two: Exploring the Free World 24 2.1. Random walk on the integers 24 2.2. Pólya’s theorem 25 2.3. Kesten’s problem 27 2.4. The classical algorithm 30 2.5. Voiculescu’s algorithm 33 2.6. Solution of Kesten’s problem 37 2.7. Spectral measures and free convolution 41 2.8. Free Poisson limit theorem 43 2.9. Semicircle flow 44 3. Lecture Three: Modelling the Free World 45 3.1. Algebraic model of a free arcsine pair 45 3.2. Algebraic model of a free semicircular pair 45 3.3. Algebraic versus asymptotic models 47 3.4. Random matrix model of a free semicircular pair 49 3.5. Random matrix model of a free pair with one semicircle 55 3.6. Random matrix model of an arbitrary free pair 57 3.7. GUE +GUE 63 3.8. GUE + deterministic 63 3.9. randomly rotated + diagonal 64 References 64
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