Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions

Baudoin, Fabrice; Wang, Jing

doi:10.1214/21-ejp600

Cited by 4 publications

(1 citation statement)

References 25 publications

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“…(1+z) M +β 1 1 0<z<∞ also defines a classical ensemble since it satisfies Pearson's equation (see for example [51]). A Brownian motion interpretation of this classical ensemble has been recently given in [10,Remark 2.5]. The corresponding ensemble is related to a Jacobi ensemble, in the sense that both weight functions are related to Euler's beta integral…”

Section: 31mentioning

confidence: 99%

Schur expansion of random-matrix reproducing kernels

Santilli,

Tierz

2021

Preprint

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We give expansions of reproducing kernels of the Christoffel-Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a q-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and q-deformed cases on the real line, we use extensions of Dotsenko-Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguerre polynomials, Painlevé tau functions and conformal block expansions is discussed in relationship to the Schur expansion obtained.

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Section: 31mentioning

confidence: 99%

Schur expansion of random-matrix reproducing kernels

Santilli,

Tierz

2021

Preprint

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Schur expansion of random-matrix reproducing kernels

Santilli¹,

Tierz²

2021

J. Phys. A: Math. Theor.

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We give expansions of reproducing kernels of the Christoffel–Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a q-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and q-deformed cases on the real line, we use extensions of Dotsenko–Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguerre polynomials, Painlevé tau functions and conformal block expansions is discussed in relationship to the Schur expansion obtained.

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A model of invariant control system using mean curvature drift from Brownian motion under submersions

Ching-Peng¹

2022

Quart. Appl. Math.

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Given a Riemannian submersion ϕ : M → N \phi : M \to N , we construct a stochastic process X X on M M such that the image Y ≔ ϕ ( X ) Y≔\phi (X) is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping π : G L ( n ) → G L ( n ) / O ( n ) \pi : GL(n) \to GL(n)/O(n) , whose image is equivalent to the space of n n -by- n n positive definite matrices, S + ( n , n ) \mathcal {S}_+(n,n) , and the said flow has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gradient of log volume of orbits. As a consequence, we are able to write down Brownian motions explicitly on several common homogeneous spaces, such as Poincaré’s upper half plane and the Bures-Wasserstein geometry on S + ( n , n ) \mathcal {S}_+(n,n) , on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson’s Brownian motion. By choosing the background metric via natural G L ( n ) GL(n) action, we arrive at an invariant control system on the G L ( n ) GL(n) -homogenous space G L ( n ) / O ( n ) GL(n)/O(n) . We investigate the feasibility of developing stochastic algorithms using the mean curvature flow.

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Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions

Cited by 4 publications

References 25 publications

Schur expansion of random-matrix reproducing kernels

Schur expansion of random-matrix reproducing kernels

Schur expansion of random-matrix reproducing kernels

A model of invariant control system using mean curvature drift from Brownian motion under submersions

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