Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices

• 成果类型: Article
• 署名作者: Maida, Mylene; Maurel-Segala, Edouard
• 署名单位: Universite Paris Saclay
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0508-x
• 发表日期: 2014
• 页码: 329-356
• 关键词: empirical measures; spectral measure; cost; CONVERGENCE

摘要

Talagrand's inequalities provide a link between two fundamentals concepts of probability: transportation and entropy. The study of the counterpart of these inequalities in the context of free probability has been initiated by Biane and Voiculescu and later extended by Hiai, Petz and Ueda for convex potentials. In this work, we prove a free analogue of a result of Bobkov and Gotze in the classical setting, thus providing free transport-entropy inequalities for a very natural class of measures appearing in random matrix theory. These inequalities are weaker than the ones of Hiai, Petz and Ueda but still hold beyond the convex case. We then use this result to get a concentration estimate for -ensembles under mild assumptions on the potential.