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The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$
- Source :
- Nonlinearity 32 (2019), 3023--3081
- Publication Year :
- 2018
-
Abstract
- The Muttalib-Borodin biorthogonal ensemble is a probability density function for $n$ particles on the positive real line that depends on a parameter $\theta$ and an external field $V$. For $\theta=\frac{1}{2}$ we find the large $n$ behavior of the associated correlation kernel with only few restrictions on $V$. The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a $3\times 3$ Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.<br />Comment: 51 pages
- Subjects :
- Mathematics - Classical Analysis and ODEs
Subjects
Details
- Database :
- arXiv
- Journal :
- Nonlinearity 32 (2019), 3023--3081
- Publication Type :
- Report
- Accession number :
- edsarx.1810.00741
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1088/1361-6544/ab247c