Indexing metadata

Hard edge tail asymptotics


 
Dublin Core PKP Metadata Items Metadata for this Document
 
1. Title Title of document Hard edge tail asymptotics
 
2. Creator Author's name, affiliation, country Jose Ramirez; Universidad de Costa Rica; Costa Rica
 
2. Creator Author's name, affiliation, country Brian Rider; CU Boulder; United States
 
2. Creator Author's name, affiliation, country Ofer Zeitouni; University of Minnesota & Weizmann Institute; United States
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random matrices, smallest singular value, hard edge
 
3. Subject Subject classification 60B20 ; 60F10
 
4. Description Abstract

Let $\Lambda$ be the limiting smallest eigenvalue in the general $(\beta,a)$-Laguerre ensemble of random matrix theory. That is, $\Lambda$ is the $n\to\infty$ distributional limit of the (scaled) minimal point drawn from the density proportional to $\Pi_1\leq i\leq j\leq n$ $$\left|\lambda_i-\lambda_j\right|^\beta\prod_{i=1}^n\lambda_i^{\frac{\beta}{2}(a+1)-1}e^{-\frac{\beta}{2}\lambda_i}$$ on $(\mathbb{R}_+^n$. Here $\beta>0$, $a> -1$; for $\beta=1,2,4$ and integer $a$, this object governs the singular values of certain rank $n$ Gaussian matrices. We prove that $$ \mathbb{P}(\Lambda>\lambda)=e^{-\frac{\beta}{2}\lambda+2\gamma\sqrt{\lambda}}\lambda^{-\frac{\gamma(\gamma+1-\beta/2)}{2\beta}} e(\beta,a)(1+o(1))$$ as $\lambda\to\infty$ in which $$\gamma = \frac{\beta}{2} (a+1)-1$$ and $e(\beta, a) > 0$ is a constant (which we do not determine). This estimate complements/extends various results previously available for special values of $\beta$ and $a$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2011-11-22
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1682
 
10. Identifier Digital Object Identifier 10.1214/ECP.v16-1682
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 16
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
15. Rights Copyright and permissions The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.

Summary of the Creative Commons Attribution License

You are free
  • to copy, distribute, display, and perform the work
  • to make derivative works
  • to make commercial use of the work
under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.