Schur–Weyl duality and the heat kernel measure on the unitary group

Lévy, Thierry

doi:10.1016/j.aim.2008.01.006

Cited by 62 publications

(79 citation statements)

References 17 publications

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“…It would be of interest to relate this limit to the approach described by Singer [12]. It should also be possible to connect the freenoise process to the large-N limit of Wilson loop expectations described in the physics literature as well as the mathematical works of Biane [1], Lévy [9], and Xu [17]. Results concerning the large-N limit of Wilson loop expectations are reviewed and developed in [11].…”

Section: Discussionmentioning

confidence: 99%

“…The large-N limit of (13) has been studied by Biane [1], Xu [17], the author [11], and, more extensively, by Lévy [9].…”

Section: Stochastic Holonomymentioning

confidence: 99%

“…On the mathematical side, Singer's paper [12] lays out several mathematical challenges in the area. More recent works by Biane [1], Lévy [9] and Xu [17] have greatly clarified the large-N limit of Wilson loop expectation values (see also [11] on this and related questions).…”

Section: Introductionmentioning

confidence: 99%

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The Large-N Yang-Mills Field on the plane and free noise

Sengupta¹

2008

AIP Conference Proceedings

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Section: Discussionmentioning

confidence: 99%

“…The large-N limit of (13) has been studied by Biane [1], Xu [17], the author [11], and, more extensively, by Lévy [9].…”

Section: Stochastic Holonomymentioning

confidence: 99%

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The Large-N Yang-Mills Field on the plane and free noise

Sengupta¹

2008

AIP Conference Proceedings

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“…This is the approach taken in [30,31], and instead, the heat kernel is evaluated at time t/N to compensate. In that sense, our limiting concentration results can be interpreted as statements about the heat kernel in a neighborhood of t = 0.…”

Section: Definition 24mentioning

confidence: 99%

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

Kemp

2015

J Theor Probab

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This paper studies the empirical laws of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups U N and the general linear groups GL N , for N ∈ N. It establishes the strongest known convergence results for the empirical eigenvalues in the U N case, and the first known almost sure convergence results for the eigenvalues and singular values in the GL N case. The limit noncommutative distribution associated with the heat kernel measure on GL N is identified as the projection of a flow on an infinite-dimensional polynomial space. These results are then strengthened from variance estimates to L p estimates for even integers p.

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“…Indeed, the one we derive below is an elaborated version of the one used in order to determine the decay order of m n (t) (see [4] p. 566):…”

Section: A Residue-type Representation Of M N (T)mentioning

confidence: 99%

Spectral Distribution of the Free Unitary Brownian Motion: Another Approach

Демни

Hmidi

2012

Lecture Notes in Mathematics

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Abstract. We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any t ∈ (0, 4) a Jordan curve γ t around the origin, not intersecting the semi-axis [1, ∞[ and whose image under some meromorphic function h t lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and h t is up to a Möbius transformation the main ingredient used in the original proof. Once we did, the spectral measure is described as the push-forward of a complex measure under a local diffeomorphism yielding its absolutecontinuity and its support. Our approach has the merit to be an easy yet technical exercise from real analysis. Reminder and MotivationIn his pioneering paper [1], Ph. Biane defined and studied the so-called free unitary or multiplicative Brownian motion. It is a unitary operator-valued Lévy process with respect to the free multiplicative convolution of probability measures on the unit circle T (or equivalently the multiplication of unitary operators that are free in some non commutative probability space). Besides, the spectral distribution µ t at any time t ≥ 0 is characterized by its momentsand m −n (t) = m n (t), n ≥ 1 since Y −1 defines a free unitary Brownian motion too. This alternate sum is not easy to handle analytically since for instance if we try to work out the moments generating function of (m n (t)) n≥1then we are led towhere in the open unit disc played a key role in the description of µ t ([2]). More precisely, it was proved there that µ t is an absolutely continuous probability measure with respect to the normalized Haar measure on T and that its density is a real analytic function inside its support. The latter coincides with T when t > 4 while it is given by the angle |θ| ≤ β(t) := (1/2) t(4 − t) + arccos(1 − (t/2)) when t ≤ 4. When proving these important results, the author relied on free stochastic integration (Lemma 11 p.266), Caratheodory's extension Theorem for Riemann maps (Lemma 12 p.270) and a Poisson-type integral representation for this kind of maps (see the proof of Proposition 10 p.270). In the present paper, we shall recover Biane's results from more simpler considerations than the ones used in the original proof. Indeed, for t ∈ (0, 4), there exists a unique piecewise smooth Jordan curve γ t around the origin, not intersecting the semi-axis [1, ∞[ and whose image under some function h t lies in T. Our construction is naturally suggested by a residue-type integral representation of m n (t) and fails when t ≥ 4. Note that the same phenomenon happens here and in Biane's proof: γ t is constructed upon two curves that have a non empty intersection if and only if t < 4, while the inverse function of τ t is defined on the interiors of two Jordan domains whose boundaries have the same phase transition ([2] p. 267). Moreover, the function h t appears in the integrand of our residue-type representation and coincides up to the Möbius transformation

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Schur–Weyl duality and the heat kernel measure on the unitary group

Cited by 62 publications

References 17 publications

The Large-N Yang-Mills Field on the plane and free noise

The Large-N Yang-Mills Field on the plane and free noise

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

Spectral Distribution of the Free Unitary Brownian Motion: Another Approach

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