Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Products of correlated symmetric matrices and q-Catalan numbers
Download PDF
Download PDF
  • Published: December 2002

Products of correlated symmetric matrices and q-Catalan numbers

  • Christian Mazza1 &
  • Didier Piau1 

Probability Theory and Related Fields volume 124, pages 574–594 (2002)Cite this article

  • 93 Accesses

  • 4 Citations

  • Metrics details

Abstract.

 The well known convergence of the spectrum of large random symmetric matrices, due to Wigner, holds for products of correlated symmetric matrices with general entries. The limiting moments coincide with weighted enumeration of permutations, or of rooted trees. When the correlations are Markovian, the limiting first moments are closely related to Carlitz-Riordan q-Catalan numbers. As a consequence, these moments asymptotically exhibit a phase transition, with respect to the correlation coefficient. The critical correlations can be computed as the least positive zero of q-hypergeometric functions. Similar methods allow to recover some results due to Logan, Mazo, Odlyzko and Shepp.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Université Claude Bernard Lyon–I, LaPCS – Domaine de Gerland, 50, avenue Tony-Garnier, 69366 Lyon Cedex 07, France. e-mail: Christian.Mazza@univ-lyon1.fr; Didier.Piau@univ-lyon1.fr, web: http://lapcs.univ-lyon1.fr, , , , , , FR

    Christian Mazza & Didier Piau

Authors
  1. Christian Mazza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Didier Piau
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 15 February 2002 / Revised version: 25 May 2002 / Published online: 24 October 2002

Mathematics Subject Classification (2000): Primary: 15A52; Secondary: 05A30, 15A18, 60F05, 82B44

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mazza, C., Piau, D. Products of correlated symmetric matrices and q-Catalan numbers. Probab Theory Relat Fields 124, 574–594 (2002). https://doi.org/10.1007/s00440-002-0225-3

Download citation

  • Issue Date: December 2002

  • DOI: https://doi.org/10.1007/s00440-002-0225-3

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Phase Transition
  • Rooted Tree
  • Symmetric Matrice
  • Weighted Enumeration
  • Positive Zero
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature