Processes on the unitary group and free probability

Abstract : This thesis focuses on the asymptotic of objects related to the Brownian motion on the unitary group in large dimension, and on the study, in free probability, of the non-commutative versions of those objects. It subdivides into essentially three parts.In Chapter 2, we solve the original problem of this thesis: the convergence of the Hall transform on the unitary group to the free Hall transform, as the dimension tends to infinity. To solve this problem, we establish theorems of existence of transition kernel for the free convolution. Finally, we use these results to prove that, exactly as the Brownian motion on the unitary group, the Brownian motion on the linear group converges in noncommutative distribution to its free version. Then we study the fluctuations around this convergence in Chapter 3. Chapter 4 presents a homomorphism between infinitely divisible measures for the free convolution, in respectively the additive case and the multiplicative case. We show that this homomorphism has a matricialversion which is based on a new model of random matrices for the free multiplicative Lévy processes.
Document type :
Theses
Complete list of metadatas

Cited literature [75 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01088217
Contributor : Abes Star <>
Submitted on : Saturday, March 7, 2015 - 2:27:00 AM
Last modification on : Wednesday, May 15, 2019 - 3:37:41 AM
Long-term archiving on : Monday, June 8, 2015 - 2:25:15 PM

File

2014PA066380.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01088217, version 2

Citation

Guillaume Cébron. Processes on the unitary group and free probability. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2014. English. ⟨NNT : 2014PA066380⟩. ⟨tel-01088217v2⟩

Share

Metrics

Record views

430

Files downloads

415