Free Brownian motion, free stochastic calculus, and random matrices

Biane, Philippe

doi:10.1090/fic/012/01

Cited by 116 publications

(277 citation statements)

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“…In particular, we have not mentioned free Brownian motion as defined in [Spe90], which appears as the limit of the Hermitian Brownian motion with size going to infinity [Bia97a]. We refer to [BiS98b] for a study of the related stochastic calculus, to [Bia98a] for the introduction of a wide class of processes with free increments and for the study of their Markov properties, to [Ans02] for the introduction of stochastic integrals with respect to processes with free increments, and to [BaNT02] for a thorough discussion of Lévy processes and Lévy laws.…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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“…Note that this equation is formally the same as the one used to construct the free unitary Brownian motion (cf. [4]). This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Monotone and boolean unitary Brownian motions

Hamdi

2015

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Abstract. The additive monotone (resp. boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (resp. boolean) independent and stationary increments which are distributed according to the arcsine law (resp. Bernoulli law) . We introduce the monotone and boolean unitary Brownian motions and we derive a closed formula for their associated moments. This provides a description of their spectral measures. We prove that, in the monotone case, the multiplicative analog of the arcsine distribution is absolutely continuous with respect to the Haar measure on the unit circle, whereas in the boolean case the multiplicative analog of the Bernoulli distribution is discrete. Finally, we use quantum stochastic calculus to provide a realization of these processes as the stochastic exponential of the correspending additive Brownian motions. IntroductionIn non-commutative probability theory, there exist several natural notions of independence. The most famous ones are the usual independence and the free independence. Other interesting examples are monotone and boolean independence. These allow to define new convolutions for probability measures. The monotone convolutions on the unit circle and the positive half-line were introduced by Bercovici in [2], see also [6]. But the additive monotone Brownian motion and the monotone Fock space where first studied by Muraki, see [9]. The monotone stochastic calculus can also be realized on the symmetric Fock space, see [5]. Bercovici studied also the boolean convolution for probability measures on the positive half-line, but it is not always defined. For probability measures on the unit circle, the boolean convolution is well defined. It was introduced by Franz in [7], see also [3]. The boolean stochastic calculus has been studied by Ben Ghorbal and Schürmann, see [1].The aim of this paper is to point out several connections between classical, free Brownian motions and their counterparts in the monotone and boolean cases. We shall consider two kinds of unitary Brownian motions, the monotone and boolean one. Both are non-commutative unitary processes, that is families of noncommutative random variables which are unitary and are characterized as having 2010 Mathematics Subject Classification. Primary 60J65, 46L51, 46L53; Secondary 65C30.

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“…While no large deviation principle is known, there exists a law of large numbers: the asymptotic distribution of the eigenvalues was computed by Biane in [4] using moment techniques. To state his results, we use the notation tr for the normalised trace on M N (C), the one such that tr(I N ) = 1.…”

Section: 2mentioning

confidence: 99%

On the Douglas-Kazakov phase transition

Lévy¹,

Maïda²

2015

ESAIM: Proc.

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Abstract. We give a rigorous proof of the fact that a phase transition discovered by Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories occurs. This phase transition can be formulated in terms of the Brownian bridge on the unitary group U(N ) when N tends to infinity. We explain how it can be understood by considering the asymptotic behaviour of the eigenvalues of the unitary Brownian bridge, and how it can be technically approached by means of Fourier analysis on the unitary group. Moreover, we advertise some more or less classical methods for solving certain minimisation problems which play a fundamental role in the study of the phase transition.

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Free Brownian motion, free stochastic calculus, and random matrices

Cited by 116 publications

References 0 publications

An Introduction to Random Matrices

An Introduction to Random Matrices

Monotone and boolean unitary Brownian motions

On the Douglas-Kazakov phase transition

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