Universality in unitary random matrix ensembles when the soft edge meets the hard edge. (English) Zbl 1147.15303
Baik, Jinho (ed.) et al., Integrable systems and random matrices. In honor of Percy Deift. Conference on integrable systems, random matrices, and applications in honor of Percy Deift’s 60th birthday, New York, NY, USA, May 22–26, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4240-9/pbk). Contemporary Mathematics 458, 265-279 (2008).
Summary: Unitary random matrix ensembles
\[ Z_{n,N}^{-1}(\det M)^\alpha\exp(-N\operatorname{Tr}V(M))\,dM \]
defined on positive definite matrices \(M\), where \(\alpha>-1\) and \(V\) is real analytic, have a hard edge at 0. The equilibrium measure associated with \(V\) typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when \(n,N\to\infty\) such that \(n/N-1={\mathcal O}(n^{-2/3})\). For each value of \(\alpha>-1\) we find a one-parameter family of limiting kernels that we describe in terms of the Hastings-McLeod solution of the Painlevé II equation with parameter \(\alpha+1/2\).
For the entire collection see [Zbl 1139.37001].
\[ Z_{n,N}^{-1}(\det M)^\alpha\exp(-N\operatorname{Tr}V(M))\,dM \]
defined on positive definite matrices \(M\), where \(\alpha>-1\) and \(V\) is real analytic, have a hard edge at 0. The equilibrium measure associated with \(V\) typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when \(n,N\to\infty\) such that \(n/N-1={\mathcal O}(n^{-2/3})\). For each value of \(\alpha>-1\) we find a one-parameter family of limiting kernels that we describe in terms of the Hastings-McLeod solution of the Painlevé II equation with parameter \(\alpha+1/2\).
For the entire collection see [Zbl 1139.37001].
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
