Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematical Physics

arXiv:1811.00064 (math-ph)
[Submitted on 31 Oct 2018 (v1), last revised 26 Jun 2019 (this version, v2)]

Title:A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions

Authors:Estelle Basor, Pavel Bleher, Robert Buckingham, Tamara Grava, Alexander Its, Elizabeth Its, Jonathan P. Keating
View PDF
Abstract:We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the sigma-Painleve V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.
Comments: 39 pages. To appear in Nonlinearity
Subjects: Mathematical Physics (math-ph); Probability (math.PR); Exactly Solvable and Integrable Systems (nlin.SI)
Cite as: arXiv:1811.00064 [math-ph]
  (or arXiv:1811.00064v2 [math-ph] for this version)
  https://doi.org/10.48550/arXiv.1811.00064
Related DOI: https://doi.org/10.1088/1361-6544/ab28c7

Submission history

From: Robert Buckingham [view email]
[v1] Wed, 31 Oct 2018 18:57:46 UTC (41 KB)
[v2] Wed, 26 Jun 2019 22:58:10 UTC (43 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
  • Other Formats
view license
Current browse context:
math-ph
< prev   |   next >
new | recent | 2018-11
Change to browse by:
math
math.MP
math.PR
nlin
nlin.SI

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)