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Time-convergent random matrices from mean-field pinned interacting eigenvalues

Time-convergent random matrices from mean-field pinned interacting eigenvalues Abstract We study a multivariate system over a finite lifespan represented by a Hermitian-valued random matrix process whose eigenvalues (i) interact in a mean-field way and (ii) converge to their weighted ensemble average at their terminal time. We prove that such a system is guaranteed to converge in time to the identity matrix that is scaled by a Gaussian random variable whose variance is inversely proportional to the dimension of the matrix. As the size of the system grows asymptotically, the eigenvalues tend to mutually independent diffusions that converge to zero at their terminal time, a Brownian bridge being the archetypal example. Unlike commonly studied random matrices that have non-colliding eigenvalues, the proposed eigenvalues of the given system here may collide. We provide the dynamics of the eigenvalue gap matrix, which is a random skew-symmetric matrix that converges in time to the $\textbf{0}$ matrix. Our framework can be applied in producing mean-field interacting counterparts of stochastic quantum reduction models for which the convergence points are determined with respect to the average state of the entire composite system. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Probability Cambridge University Press

Time-convergent random matrices from mean-field pinned interacting eigenvalues

Mengütürk, Levent Ali
Journal of Applied Probability , Volume 60 (2): 24 – Jun 1, 2023
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Publisher
Cambridge University Press
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust
ISSN
1475-6072
eISSN
0021-9002
DOI
10.1017/jpr.2022.53
Publisher site
See Article on Publisher Site

Abstract

Abstract We study a multivariate system over a finite lifespan represented by a Hermitian-valued random matrix process whose eigenvalues (i) interact in a mean-field way and (ii) converge to their weighted ensemble average at their terminal time. We prove that such a system is guaranteed to converge in time to the identity matrix that is scaled by a Gaussian random variable whose variance is inversely proportional to the dimension of the matrix. As the size of the system grows asymptotically, the eigenvalues tend to mutually independent diffusions that converge to zero at their terminal time, a Brownian bridge being the archetypal example. Unlike commonly studied random matrices that have non-colliding eigenvalues, the proposed eigenvalues of the given system here may collide. We provide the dynamics of the eigenvalue gap matrix, which is a random skew-symmetric matrix that converges in time to the $\textbf{0}$ matrix. Our framework can be applied in producing mean-field interacting counterparts of stochastic quantum reduction models for which the convergence points are determined with respect to the average state of the entire composite system.

Journal

Journal of Applied Probability – Cambridge University Press

Published: Jun 1, 2023

Keywords: Mean-field interacting systems; pinned diffusions; quantum reduction; random matrix; 60G; 60H

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