Probability Theory and Related Fields

, Volume 124, Issue 4, pp 574-594

First online:

Products of correlated symmetric matrices and q-Catalan numbers

  • Christian MazzaAffiliated withUniversité 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
  • , Didier PiauAffiliated withUniversité 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

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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.