### Abstract

Voiculescu's single variable free entropy is generalized in two different ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. Next, the perturbation theory for compactly supported probability measures via free relative entropy is developed on the analogy of the perturbation theory via relative entropy. When the perturbed measure via relative entropy is suitably arranged on the space of selfadjoint matrices and the matrix size goes to infinity, it is proven that the perturbation via relative entropy on the matrix space approaches asymptotically to that via free relative entropy. The whole theory can be adapted to probability measures on the unit circle.

Original language | English |
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Pages (from-to) | 679-718 |

Number of pages | 40 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 54 |

Issue number | 3 |

Publication status | Published - Jul 2002 |

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### Keywords

- Free entropy
- Free relative entropy
- Large deviation
- Random matrix

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the Mathematical Society of Japan*,

*54*(3), 679-718.

**Free relative entropy for measures and a corresponding perturbation theory.** / Hiai, Fumio; Mizuo, Masaru; Petz, D.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 54, no. 3, pp. 679-718.

}

TY - JOUR

T1 - Free relative entropy for measures and a corresponding perturbation theory

AU - Hiai, Fumio

AU - Mizuo, Masaru

AU - Petz, D.

PY - 2002/7

Y1 - 2002/7

N2 - Voiculescu's single variable free entropy is generalized in two different ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. Next, the perturbation theory for compactly supported probability measures via free relative entropy is developed on the analogy of the perturbation theory via relative entropy. When the perturbed measure via relative entropy is suitably arranged on the space of selfadjoint matrices and the matrix size goes to infinity, it is proven that the perturbation via relative entropy on the matrix space approaches asymptotically to that via free relative entropy. The whole theory can be adapted to probability measures on the unit circle.

AB - Voiculescu's single variable free entropy is generalized in two different ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. Next, the perturbation theory for compactly supported probability measures via free relative entropy is developed on the analogy of the perturbation theory via relative entropy. When the perturbed measure via relative entropy is suitably arranged on the space of selfadjoint matrices and the matrix size goes to infinity, it is proven that the perturbation via relative entropy on the matrix space approaches asymptotically to that via free relative entropy. The whole theory can be adapted to probability measures on the unit circle.

KW - Free entropy

KW - Free relative entropy

KW - Large deviation

KW - Random matrix

UR - http://www.scopus.com/inward/record.url?scp=0036629981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036629981&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036629981

VL - 54

SP - 679

EP - 718

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -