On Classical Analogues of Free Entropy Dimension
نویسندگان
چکیده
Abstract. We define a classical probability analog of Voiculescu’s free entropy dimension that we shall call the classical probability entropy dimension. We show that the classical probability entropy dimension is related with diverse other notions of dimension. First, it equals the fractal dimension. Second, if one extends Bochner’s inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner’s inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner’s inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.
منابع مشابه
Notes on Free Probability Theory
Lecture 1. Free Independence and Free Harmonic Analysis. 2 1.1. Probability spaces. 2 1.2. Non-commutative probability spaces. 3 1.3. Classical independence. 4 1.4. Free products of non-commutative probability spaces. 5 1.5. Free Fock space. 6 1.6. Free Central Limit Theorem. 8 1.7. Free Harmonic Analysis. 10 1.8. Further topics. 16 Lecture 2. Random Matrices and Free Probability. 17 2.1. Rando...
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