Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918047266111488 |
|---|---|
| author | Deaño, Alfredo McLaughlin, Kenneth T-R Molag, Leslie Simm, Nick |
| author_facet | Deaño, Alfredo McLaughlin, Kenneth T-R Molag, Leslie Simm, Nick |
| contents | We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} dμ(z) = (1-|z|^{2})^{α-1}|z-x|^γ\mathbf{1}_{|z| < 1}d^{2}z, \end{equation*} where $d^{2}z$ is the two dimensional area measure, $α$ is a parameter that can grow with $n$, while $γ>-2$ and $x>0$ are fixed. This measure arises naturally in the study of characteristic polynomials of non-Hermitian ensembles and generalises the example of a Gaussian weight that was recently studied by several authors. We obtain asymptotics in all regions of the complex plane and via an appropriate differential identity, we obtain the asymptotic expansion of the partition function. The main approach is to convert the planar orthogonality to one defined on suitable contours in the complex plane. Then the asymptotic analysis is performed using the Deift-Zhou steepest descent method for the associated Riemann-Hilbert problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices Deaño, Alfredo McLaughlin, Kenneth T-R Molag, Leslie Simm, Nick Mathematical Physics Classical Analysis and ODEs Complex Variables Probability 33C45, 33E17, 60B20, 41A60 We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} dμ(z) = (1-|z|^{2})^{α-1}|z-x|^γ\mathbf{1}_{|z| < 1}d^{2}z, \end{equation*} where $d^{2}z$ is the two dimensional area measure, $α$ is a parameter that can grow with $n$, while $γ>-2$ and $x>0$ are fixed. This measure arises naturally in the study of characteristic polynomials of non-Hermitian ensembles and generalises the example of a Gaussian weight that was recently studied by several authors. We obtain asymptotics in all regions of the complex plane and via an appropriate differential identity, we obtain the asymptotic expansion of the partition function. The main approach is to convert the planar orthogonality to one defined on suitable contours in the complex plane. Then the asymptotic analysis is performed using the Deift-Zhou steepest descent method for the associated Riemann-Hilbert problem. |
| title | Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices |
| topic | Mathematical Physics Classical Analysis and ODEs Complex Variables Probability 33C45, 33E17, 60B20, 41A60 |
| url | https://arxiv.org/abs/2505.12633 |