Products of correlated symmetric matrices and q-Catalan numbers

• 成果类型: Article
• 署名作者: Mazza, C; Piau, D
• 署名单位: Universite Claude Bernard Lyon 1
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-002-0225-3
• 发表日期: 2002
• 页码: 574-594
• 关键词: characteristic vectors; bordered matrices

摘要

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.