Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability
| Autor: | David Jekel |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Free entropy
Mathematics::Commutative Algebra General Mathematics Probability (math.PR) 010102 general mathematics Mathematics - Operator Algebras Regular polygon Condensed Matter::Mesoscopic Systems and Quantum Hall Effect 16. Peace & justice Free probability Conditional expectation 01 natural sciences 010104 statistics & probability Primary: 46L54 Secondary: 35K10 37A35 46L52 46L53 60B20 Law FOS: Mathematics 0101 mathematics Operator Algebras (math.OA) Random matrix Random variable Mathematics - Probability Corresponding conditional Mathematics |
| Zdroj: | International Mathematics Research Notices. 2022:4514-4619 |
| ISSN: | 1687-0247 1073-7928 |
| Popis: | Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, \dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots,X_m) \to \mathrm{W}^*(S_1,\dots,S_m)$ which maps $\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(S_1,\dots,S_k)$ for each $k = 1, \dots, m$, and which also witnesses the Talagrand inequality for the law of $(X_1,\dots,X_m)$ relative to the law of $(S_1,\dots,S_m)$. 73 pages |
| Databáze: | OpenAIRE |
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