Cumulants for random matrices as convolutions on the symmetric group

Probability Theory and Related Fields

Volumn 136, Issue 1, 2006, Pages 19-36

  Capitaine, M ^a   Casalis, M ^b  

^a CNRS   (France)

^b UNIVERSITÉ PAUL SABATIER   (France)

Author keywords

Cumulants; Free probability; Matricial moments; Random matrices

Indexed keywords

EID: 33745457926     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-005-0474-z     Document Type: Review

References (21)

1. Biane, Ph.: Representations of symmetric groups and free probability. Adv. Math. 138, 126-181 (1998)
2. Biane, Ph.: Minimal Factorizations of a Cycle and Central Multiplicative Functions on the Infinite Symmetric Group. Journal Comb. Theory, Series A, 76, 197-212 (1996)
3. Capitaine, M., Casalis, M.: Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to Beta random matrices. Indiana University Mathematics Journal, Vol. 53 (2), 397-431 (2004)
4. Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. Int. Math.res. Not. 17, 953-982 (2003)
5. Collins, B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. To appear in Probability Theory and Related Fields
6. Collins, B., Sniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. arXiv: math-ph/0402073 v1
7. Dykema, K.J.: On certain free product factors via an extended matrix model. J. Funct. Anal. 112, 31-60 (1993)
8. Graczyk, P., Letac, G., Massam, H.: The complex Wishart distribution and the symmetric group. Annals of Statistics, 31, 287-309 (2003)
9. Graczyk P., Letac G., Massam H.: Hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution. J. Theoret. Proba, To appear
10. Lehner, F.: Free Cumulants and Enumeration of Connected Partitions. Europ. J. Combinatorics, 23, 1025-1031 (2002)
11. Lehner, F.: Cumulants in noncommutative probability theory I, Non commutative exchangeability systems. Math. Zeitschrift, 28 (1), 67-100 (2004)
12. Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs Volume 77, Amer. Math. Soc.
13. Krawczyk, B., Speicher, R.: Combinatorics of free cumulants. J. Combin. Theory Ser. A, 90, 267-292 (2000)
14. Mingo, J.A., Nica, A.: Annular Noncrossing Permutations and Partitions and Second Order Asymptotics for Random Matrices. IMRN, 28, 1413-1460 (2004)
15. Mingo, J.A., Speicher, R.: Second order freeness and fluctuations of random matrices: I: Gaussian and Wishart matrices and cyclic Fock Spaces. arXiv: math.OA/0405191 v2
16. Mingo, J.A., Sniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices: II: Unitary random matrices. arXiv: math.OA/0405258 v1
17. Nica, A., Speicher, R.: On the multiplication of free N-uplss of noncommutative random variables. Am. Journ. of Math. 118, 799-837 (1996)
18. Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math Ann. 298, 611-628 (1994)
19. Voiculescu, D.V.: Addition of certain non-commuting random variables J. Funct. Anal. 66, 323-346 (1986)
20. Voiculescu, D.V.: Limit laws for random matrices and free products. Invent. Math. 104, 201-220 (1991)
21. Xu, R: A random matrix model from 2-dimensional Yang-Mills theory. Comm. Math. Phys. 190, 287-307 (1997)