On classical analogues of free entropy dimension |
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Authors: | A Guionnet D Shlyakhtenko |
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Institution: | a École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, UMR 5669, 46 Allée d'Italie, 69364 Lyon Cedex 07, France b The Miller institute for Basic Research in Science, University of California, Berkeley, CA, USA c Department of Mathematics, UCLA, Los Angeles, CA 90095, USA |
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Abstract: | We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of 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. |
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Keywords: | Free probability Free entropy dimension Fractal dimension Measure Bochner inequality |
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