Probability Theory and Related Fields

, Volume 124, Issue 4, pp 574–594

Products of correlated symmetric matrices and q-Catalan numbers

  • Christian Mazza
  • Didier Piau

DOI: 10.1007/s00440-002-0225-3

Cite this article as:
Mazza, C. & Piau, D. Probab Theory Relat Fields (2002) 124: 574. doi:10.1007/s00440-002-0225-3

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Christian Mazza
    • 1
  • Didier Piau
    • 1
  1. 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.frFR