scholarly journals On the local semicircular law for Wigner ensembles

Bernoulli ◽  
2018 ◽  
Vol 24 (3) ◽  
pp. 2358-2400 ◽  
Author(s):  
Friedrich Götze ◽  
Alexey Naumov ◽  
Alexander Tikhomirov ◽  
Dmitry Timushev
Keyword(s):  
Semicircular Law

scholarly journals Cumulants of the q-semicircular law, Tutte polynomials, and heaps

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Matthieu Josuat-Vergès
Keyword(s):  
Tutte Polynomial ◽  
Combinatorial Interpretation ◽  
Poisson Law ◽  
Semicircular Law ◽  
Combinatorial Model ◽  
Tutte Polynomials ◽  
International Audience ◽  
Nonnegative Coefficients

International audience The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law. La loi q-semicirculaire introduite par Bożejko et Speicher interpole entre la loi gaussienne et la loi semi-circulaire, et ses moments ont une interprétation combinatoire en termes de couplages et croisements. Nous prouvons que les cumulants de cette loi sont, à un facteur près, des polynômes en q à coefficients positifs. La méthode consiste à obtenir ces cumulants par une énumération de couplages connexes, pondérés par l’évaluation en (1,q) d'un polynôme de Tutte. Les cas particuliers q=0 et q=2 ont une preuve alternative, reliè au fait que des évaluations particulières du polynôme de Tutte comptent certaines orientations de graphes. Nos méthodes donnent aussi un modèle combinatoire aux cumulants de la loi de Poisson libre.

scholarly journals On the Semicircular Law of Large-Dimensional Random Quaternion Matrices

2015 ◽  
Vol 29 (3) ◽  
pp. 1100-1120 ◽  
Author(s):  
Yanqing Yin ◽  
Zhidong Bai ◽  
Jiang Hu
Keyword(s):  
Quaternion Matrices ◽  
Semicircular Law
Sign in / Sign up

Export Citation Format

Share Document