Skip to main content
SpringerLink
Log in
Book cover

Introduction to Random Matrices pp 67–74Cite as

Finite N

  • Chapter
  • First Online:

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 26))

Abstract

Look back at Chap. 1, where we constructed Gaussian matrices and histogrammed their eigenvalues.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Hereafter, inside a determinant the indices of the entries will run from 1 to N.

  2. 2.

    Use \(\left( \prod _\ell \alpha _\ell \right) \det (f(i,j))=\det (\sqrt{\alpha _i\alpha _j}f(i,j))\).

  3. 3.

    Note that there is a factor (1 / 2) multiplying \(V(x)\) in the kernel (10.7), while there is none in the weight function of the orthonormal polynomials in (10.8).

  4. 4.

    The most accurate reference seems however to be [3].

  5. 5.

    What does in the bulk mean? The point is that Hermite polynomials (and other classical orthogonal polynomials) have two different asymptotics, according to the way their argument and parameter scale with N. This in turns corresponds to different regimes, namely different locations \(x\) where the spectrum is looked at, and different zooming resolutions.

References

  1. Y.V. Fyodorov, Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond (2004), https://arxiv.org/pdf/math-ph/0412017.pdf

  2. E. Kanzieper, G. Akemann, Phys. Rev. Lett. 95, 230201 (2005); G. Akemann, E. Kanzieper, J. Stat. Phys. 129, 1159 (2007)

    Google Scholar 

  3. G. Mahoux, M.L. Mehta, J. Phys. I France 1, 1093 (1991)

    Article  Google Scholar 

  4. G. Akemann, L. Shifrin, J. Phys. A : Math. Gen. 40, F785 (2007)

    Article  ADS  Google Scholar 

  5. W. Van Assche, Asymptotics for orthogonal polynomials (Springer, 2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University College London, London, UK

    Giacomo Livan

  2. Instituto de Física, Universidade Federal de Uberlândia, Uberlândia, Minas Gerais, Brazil

    Marcel Novaes

  3. Department of Mathematics, King’s College London, London, UK

    Pierpaolo Vivo

Authors
  1. Giacomo Livan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcel Novaes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pierpaolo Vivo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierpaolo Vivo .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 The Author(s)

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Livan, G., Novaes, M., Vivo, P. (2018). Finite N . In: Introduction to Random Matrices. SpringerBriefs in Mathematical Physics, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-70885-0_10

Download citation

Publish with us

Policies and ethics

Search

Navigation