Skip to main content

Large deviations for the law of the spectral measure of Gaussian Wigner's matrices

  • 1661 Accesses

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1957)

Abstract

In this section, we consider the law of N random variables (?1, . . . , ?N) with

$$P_{V,\beta }^N (d\lambda _1 , \ldots ,d\lambda _N ) = (Z_{V,\beta }^N )^{ - 1} |\Delta (\lambda )|^\beta e - N\sum\nolimits_{i = 1}^N {V(\lambda _i )\prod\limits_{i = 1}^N {d\lambda _i } ,} $$
((10.1))

for a continuous function V : R ? R such that

$$\mathop {\lim {\rm inf}}\limits_{|x| \to \infty } \frac{{V(x)}}{{\beta \log |x|}} > 1$$
((10.2))

and a positive real number ?. Here, ?(?) = ?1?i<j?N(?i ? ?j ).

When V (x) = 4-1?x2, we have seen in Lemma IV that \(P_{4^{ - 1} \beta x^2 ,\beta }^N \) is the law of the eigenvalues of an N ×N GOE (resp. GUE, resp GSE) matrix when ? = 1 (resp. ? = 2, resp. ? = 4). The case ? = 4 corresponds to another matrix ensemble, namely the GSE. In view of these remarks and other applications discussed in Part III, we consider in this section the slightly more general model with a potential V . We emphasize, however, that the distribution (10.1) precludes us from considering random matrices with independent non-Gaussian entries.

Keywords

  • Spectral Measure
  • Large Deviation Principle
  • Concentration Inequality
  • Monotone Convergence Theorem
  • Bibliographical Note

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (Canada)
  • 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alice Guionnet .

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Guionnet, A. (2009). Large deviations for the law of the spectral measure of Gaussian Wigner's matrices. In: Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics(), vol 1957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69897-5_11

Download citation