What is the probability that a large random matrix has no real Eigenvalues?
| Autor: | Eugene Kanzieper, Carsten Timm, Roger Tribe, Mihail Poplavskyi, Oleg Zaboronski |
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| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
High Energy Physics - Theory FOS: Physical sciences 01 natural sciences large deviations Matrix (mathematics) 0103 physical sciences FOS: Mathematics Limit (mathematics) 0101 mathematics 010306 general physics QA Eigenvalues and eigenvectors Mathematical Physics Mathematics Real Ginibre ensemble 60B20 Nonlinear Sciences - Exactly Solvable and Integrable Systems 010102 general mathematics Mathematical analysis Probability (math.PR) Mathematical Physics (math-ph) High Energy Physics - Theory (hep-th) Large deviations theory Statistics Probability and Uncertainty Exactly Solvable and Integrable Systems (nlin.SI) Random matrix Mathematics - Probability 60F10 |
| Zdroj: | Ann. Appl. Probab. 26, no. 5 (2016), 2733-2753 |
| ISSN: | 1050-5164 |
| Popis: | We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2n\times 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k}=\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,0}= -\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ where $\zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{n\geq 1}$, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k_n}=-\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ provided $\lim_{n\rightarrow \infty} \left(n^{-1/2}\log(n)\right) k_{n}=0$. Comment: 23 pages, 1 figure |
| Databáze: | OpenAIRE |
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