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Dec. 6th, 2023

Random Interacting Systems, Scaling Limits, and Universality

Department of Physics, Chuo University, Tokyo

Saori MORIMOTO Joint work with M.Katori and T.Shirai

Eigenvalue and pseudospectrum processes generated by the shift matrix with perturbation

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Outline

  • Introduction
  • Two kinds of the matrix-valued BM with a matrix
  • Shift matrix and dynamics with Perturbation
  • Theorems on the eigenvalue processes
  • Definition of pseudospectrum
  • Eigenvalues and pseudospectrum
  • Summary
  • Future problem

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Introduction

 

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The matrix-valued BM starts from the null matrix

 

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The matrix-valued BM starts from the shift matrix

 

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Shift matrix with Gaussian random perturbation

 

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Shift dynamics with Gaussian random perturbation

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Shift dynamics with uniform perturbation

 

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Theorems on the eigenvalue processes

 

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Theorem 1

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Theorems on the eigenvalue processes

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Corollary 2

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Theorems on the eigenvalue processes

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Definition of pseudospectrum

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Eigenvalues and pseudospectrum

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Eigenvalues and pseudospectrum

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Eigenvalues and pseudospectrum

 

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Summary

 

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Future problems

  1. Numerical calculation suggests eigenvalue-like (peak) structures in the pseudospectrum. Analytic study is desired.

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eigenvalues

pseudospectrum

eigenvalue-like (peak) structures

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Future problems

 

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Lloyd N. Trefethen, Mark Embree, `Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators', Princeton University Press, Princeton and Oxford, 2005

Theorem [ Trefethen-Embree 2005 ]

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Future problems

 

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Future problems

 

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References

[1] Burda, Z., Grela, J., Nowak, M. A., Tarnowski, W., Warchol, P.: Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics. Nucl. Phys. B 897, 421-447 (2015)

[2] Trefethen, L. N., Embree, M.: Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators, vol. 1. Princeton University Press, Princeton (2005)

[3] Lloyd N. Trefethen, Mark Embree, `Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators', Princeton University Press, Princeton and Oxford, 2005

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Thank you very much for your attention.