Second order cumulants of products
Authors:
James A. Mingo, Roland Speicher and Edward Tan
Journal:
Trans. Amer. Math. Soc. 361 (2009), 4751-4781
MSC (2000):
Primary 46L54; Secondary 15A52, 60F05
DOI:
https://doi.org/10.1090/S0002-9947-09-04696-0
Published electronically:
April 21, 2009
MathSciNet review:
2506426
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of Krawczyk and Speicher. We apply our result to the problem of calculating the second order cumulants of a semi-circular and Haar unitary operator.
- 1. J. Ambjørn, J. Jurkiewicz, and Yu. M. Makeenko, Multiloop correlators for two-dimensional quantum gravity, Phys. Lett. B 251 (1990), no. 4, 517–524. MR 1084539, https://doi.org/10.1016/0370-2693(90)90790-D
- 2. Greg W. Anderson and Ofer Zeitouni, A CLT for a band matrix model, Probab. Theory Related Fields 134 (2006), no. 2, 283–338. MR 2222385, https://doi.org/10.1007/s00440-004-0422-3
- 3. Z. D. Bai and Jack W. Silverstein, CLT for linear spectral statistics of large-dimensional sample covariance matrices, Ann. Probab. 32 (2004), no. 1A, 553–605. MR 2040792, https://doi.org/10.1214/aop/1078415845
- 4. Philippe Biane, Some properties of crossings and partitions, Discrete Math. 175 (1997), no. 1-3, 41–53. MR 1475837, https://doi.org/10.1016/S0012-365X(96)00139-2
- 5. É. Brézin and A. Zee, Universality of the correlations between eigenvalues of large random matrices, Nuclear Phys. B 402 (1993), no. 3, 613–627. MR 1236191, https://doi.org/10.1016/0550-3213(93)90121-5
- 6. Benoît Collins, James A. Mingo, Piotr Śniady, and Roland Speicher, Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants, Doc. Math. 12 (2007), 1–70. MR 2302524
- 7. P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1677884
- 8. Persi Diaconis, Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 2, 155–178. MR 1962294, https://doi.org/10.1090/S0273-0979-03-00975-3
- 9. Persi Diaconis and Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49–62. Studies in applied probability. MR 1274717, https://doi.org/10.2307/3214948
- 10. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
- 11. Fumio Hiai and Dénes Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000. MR 1746976
- 12. Kurt Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), no. 1, 151–204. MR 1487983, https://doi.org/10.1215/S0012-7094-98-09108-6
- 13. V. F. R. Jones, A quotient of the affine Hecke algebra in the Brauer algebra, Enseign. Math. (2) 40 (1994), no. 3-4, 313–344. MR 1309131
- 14. A. Khorunzhy, B. Khoruzhenko, and L. Pastur, On the 1/𝑁 corrections to the Green functions of random matrices with independent entries, J. Phys. A 28 (1995), no. 1, L31–L35. MR 1325832
- 15. C. King, Two-dimensional Potts model and annular partitions, J. Statist. Phys. 96 (1999), no. 5-6, 1071–1089. MR 1722987, https://doi.org/10.1023/A:1004692301344
- 16. Bernadette Krawczyk and Roland Speicher, Combinatorics of free cumulants, J. Combin. Theory Ser. A 90 (2000), no. 2, 267–292. MR 1757277, https://doi.org/10.1006/jcta.1999.3032
- 17. G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), no. 4, 333–350 (French). MR 309747, https://doi.org/10.1016/0012-365X(72)90041-6
- 18. Timothy Kusalik, James A. Mingo, and Roland Speicher, Orthogonal polynomials and fluctuations of random matrices, J. Reine Angew. Math. 604 (2007), 1–46. MR 2320312, https://doi.org/10.1515/CRELLE.2007.018
- 19. V. P. Leonov and A. N. Sirjaev, On a method of semi-invariants, Theor. Probability Appl. 4 (1959), 319–329. MR 123345, https://doi.org/10.1137/1104031
- 20. James A. Mingo and Alexandru Nica, Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices, Int. Math. Res. Not. 28 (2004), 1413–1460. MR 2052516, https://doi.org/10.1155/S1073792804133023
- 21. James A. Mingo, Piotr Śniady, and Roland Speicher, Second order freeness and fluctuations of random matrices. II. Unitary random matrices, Adv. Math. 209 (2007), no. 1, 212–240. MR 2294222, https://doi.org/10.1016/j.aim.2006.05.003
- 22. James A. Mingo and Roland Speicher, Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces, J. Funct. Anal. 235 (2006), no. 1, 226–270. MR 2216446, https://doi.org/10.1016/j.jfa.2005.10.007
- 23. Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879
- 24.
R. Rao, J. Mingo, A. Edelman, and R. Speicher,
Statistical eigen-inference from large Wishart matrices, arXiv:math/0701314 - 25. Roland Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994), no. 4, 611–628. MR 1268597, https://doi.org/10.1007/BF01459754
- 26. W. T. Tutte, A census of slicings, Canadian J. Math. 14 (1962), 708–722. MR 142470, https://doi.org/10.4153/CJM-1962-061-1
- 27. W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 (1968), 64–74. MR 218276, https://doi.org/10.1090/S0002-9904-1968-11877-4
- 28. Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. MR 839105, https://doi.org/10.1016/0022-1236(86)90062-5
- 29. D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253
corrections to the Green functions of random matrices with independent entries, J. Phys. A 28 (1995), L31-L35. Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L54, 15A52, 60F05
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Additional Information
James A. Mingo
Affiliation:
Department of Mathematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Email:
mingo@mast.queensu.ca
Roland Speicher
Affiliation:
Department of Mathematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Email:
speicher@mast.queensu.ca
Edward Tan
Affiliation:
Department of Mathematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Email:
3et8@qlink.queensu.ca
DOI:
https://doi.org/10.1090/S0002-9947-09-04696-0
Received by editor(s):
August 17, 2007
Published electronically:
April 21, 2009
Additional Notes:
The research of the first and second authors was supported by Discovery Grants and a Leadership Support Initiative Award from the Natural Sciences and Engineering Research Council of Canada
The research of the second author was supported by a Killam Fellowship from the Canada Council for the Arts.
The research of the third author was supported by an Undergraduate Student Research Award from the Natural Sciences and Engineering Research Council of Canada
Article copyright:
© Copyright 2009
American Mathematical Society
