Stein's method and random character ratios

Transactions of the American Mathematical Society

Volumn 360, Issue 7, 2008, Pages 3687-3730

  Fulman, Jason ^a,b  

^a UNIVERSITY OF PITTSBURGH   (United States)

^b UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Association scheme; Character ratio; Gelfand pair; Normal approximation; Plancherel measure; Stein's method; Symmetric group

Indexed keywords

EID: 57649132751     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-08-04371-7     Document Type: Article

References (45)

1. Aldous, D. and Diaconis, P., Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), 413-432. MR1694204 (2000g:60013)
2. Bannai, E. and Ito, T., Algebraic combinatorics, I. Association schemes, Benjamin/ Cummings Publishing Co., Inc., Menlo Park, CA, 1984. MR882540 (87m:05001)
3. Borodin, A., Okounkov, A., and Olshanski, G., Asymptotics of Plancherel measure for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481-515. MR1758751 (2001g:05103)
4. Borodin, A. and Olshanski, G., Z-measures on partitions and their scaling limits, European J. Combin. 26 (2005), 795-834. MR2143199 (2006d:60018)
5. Borodin, A. and Olshanski, G., Harmonic functions on multiplicative graphs and interpolation polynomials, Elecron. J. Combin. 7 (2000), Research paper 28, 39 pages. MR1758654 (2001f:05160)
6. Chatterjee, S. and Fulman, J., Exponential approximation by exchangeable pairs and spectral graph theory, (2006), preprint math.PR/0605552 at http://xxx.lanl.gov.
7. Deift, P., Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), 631-640. MR1764262 (2001g:05012)
8. Diaconis, P., Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes, Volume 11, 1988. MR964069 (90a:60001)
9. Diaconis, P. and Holmes, S., Random walks on trees and matchings, Elec. J. Probab. 7 (2002), 17 pages (electronic). MR1887626 (2002k:60025)
10. Diaconis, P. and Shahshahani, M., Generating a random permutation with random transpositions, Z. Wahr. Verw. Gebiete 57 (1981), 159-179. MR626813 (82h:60024)
11. Durrett, R., Probability: Theory and examples, Second edition, Duxbury Press, Belmont, CA, 1996. MR1609153 (98m:60001)
12. Eskin, A. and Okounkov, A., Asymptotics of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001), 59-103. MR1839286 (2002g:32018) (Pubitemid 33725743)
13. Eymard, P. and Roynette, B., Marches aléatoires sur le dual de SU(2), in Analyse harmonique sur les groupes de Lie, Springer Lecture Notes in Math, Volume 497 (1975), 108-152. MR0423457 (54:11435)
14. Fulman, J., Stein's method and Plancherel measure of the symmetric group, Transac. Amer. Math. Soc. 357 (2005), 555-570. MR2095623 (2005e:05156)
15. Fulman, J., Stein's method, Jack measure, and the Metropolis algorithm, J. Combin. Theory Ser. A 108 (2004), 275-296. MR2098845 (2005h:60021)
16. Fulman, J., An inductive proof of the Berry-Esseen theorem for character ratios, Ann. Combin. 10 (2006), 319-332. MR2284273 (2007i:05189)
17. Fulman, J., Martingales and character ratios, Trans. Amer. Math. Soc. 358 (2006), 4533-4552. MR2231387 (2007e:05178)
18. Gross, B., Some applications of Gelfand pairs to number theory, Bull. Amer. Math. Soc. 24 (1991), 277-301. MR1074028 (91i:11055)
19. Hanlon, P., Stanley, R., and Stembridge, J., Some combinatorial aspects of the spectra of normally distributed random matrices, in Hypergeometric functions on domains of positivity, Jack polynomials, and applications, Contemporary Mathematics, Volume 138 (1992), 151-174. MR1199126 (93j:05164)
20. Hoffman, P. and Humphreys, J., Projective representations of the symmetric group, Oxford University Press, New York, 1992. MR1205350 (94f:20027)
21. Hora, A., Central limit theorem for the adjacency operators on the infinite symmetric group, Comm. Math. Phys. 195 (1998), 405-416. MR1637801 (99i:46058)
22. Hora, A., Central limit theorems and asymptotic spectral analysis in large graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 221-246. MR1628244 (99g:46089)
23. Ivanov, V., The Gaussian limit for projective characters of large symmetric groups, J. Math. Sci. (N.Y.), 121 (2004), 2330-2344. Translated from Zap. Nauchn. Sem. POMI 283 (2001), 73-97. MR1879064 (2002k:60061)
24. Ivanov, V. and Olshanski, G., Kerov's central limit theorem for the Plancherel measure on Young diagrams, in Symmetric Functions 2001: Surveys of developments and perspectives, Kluwer Academic Publishers, Dodrecht, 2002.
25. Johansson, K., Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. 153 (2001), 259-296. MR1826414 (2002g:05188)
26. Kerov, S.V., Gaussian limit for the Plancherel measure of the symmetric group, Compt. Rend. Acad. Sci. Paris, Serie I 316 (1993), 303-308. MR1204294 (93k:20106)
27. Kerov, S.V., Anisotropic Young diagrams and Jack symmetric functions, Funct. Anal. Appl. 34 (2000), 41-51. MR1756734 (2001f:05158)
28. Letac, G., Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov, Adv. Appl. Probab. 14 (1982), 272-294. MR650123 (83i:60078)
29. Macdonald, I., Symmetric functions and Hall polynomials, Second edition, Oxford University Press, New York, 1995. MR1354144 (96h:05207)
30. Macwilliams, F. and Sloane, N., The theory of error-correcting codes, Elsevier Science B.V., Amsterdam, 1977.
31. Matsumoto, S., Correlation functions of the shifted Schur measure, J. Math. Soc. Japan 57 (2005), 619-637. MR2139724 (2006a:60013)
32. Okounkov, A., Random matrices and random permutations, Internat. Math. Res. Notices 20 (2000), 1043-1095. MR1802530 (2002c:15045)
33. Okounkov, A., The uses of random partitions, XIVth International Congress on Mathematical Physics, 379-403, World Sci. Publ., Hackensack, NJ 2005. MR2227852 (2007c:05013)
34. Reinert, G., Couplings for normal approximations with Stein's method, in Microsurveys in discrete probability, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Volume 41 (1998), 193-207. MR1630415 (99h:60043)
35. Rinott, Y. and Rotar, V., On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics, Ann. of Appl. Probab. 7 (1997), 1080-1105. MR1484798 (99g:60050)
36. Rinott, Y. and Rotar, V., Normal approximations by Stein's method, Decis. Econ. Finance 23 (2000), 15-29. MR1780090 (2001h:60037)
37. Sagan, B., The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Second edition, Springer-Verlag, New York, 2001. MR1824028 (2001m:05261)
38. Serre, J-P., Linear representations of finite groups, Springer-Verlag, New York, 1977. MR0450380 (56:8675)
39. Shao, Q., and Su, Z., The Berry-Esseen bound for character ratios, Proc. Amer. Math. Soc. 134 (2006), 2153-2159. MR2215787
40. Sniady, P., Gaussian fluctuations of characters of symmetric groups and of Young diagrams, Probab. Theory Related Fields 136 (2006), 263-297. MR2240789 (2007d:20020)
41. Stein, C., Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes, Volume 7, 1986. MR882007 (88j:60055)
42. Stembridge, J., On Schur's Q-functions and the primitive idempotents of a commutative Hecke algebra, J. Algebraic Combin. 1 (1992), 71-95. MR1162642 (93c:05113)
43. Terras, A., Fourier analysis on finite groups and applications, London Math. Society Student Texts 43, Cambridge University Press, Cambridge, 1999. MR1695775 (2000d:11003)
44. Terras, A., Survey of spectra of Laplacians on finite symmetric spaces, Experiment. Math. 5 (1996), 15-32. MR1412951 (97m:11154)
45. Tracy, C. and Widom, H., A limit theorem for shifted Schur measures, Duke Math. Journal 123 (2004), 171-208. MR2060026 (2006c:60028)