$\mathfrak{sl}(2)$ operators and Markov processes on branching graphs
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- 作者：Leonid Petrov
- 关键词：Linear operators on graphs ; Markov processes on branching graphs ; Young graph ; z ; Measures on partitions ; Meixner symmetric functions ; Ewens–Pitman’s partition structures
- 刊名：Journal of Algebraic Combinatorics
- 出版年：2013
- 出版时间：November 2013
- 年：2013
- 卷：38
- 期：3
- 页码：663-720
- 全文大小：1770KB
- 参考文献：1. Aissen, M., Edrei, A., Schoenberg, I.J., Whitney, A.: On the generating functions of totally positive sequences. Proc. Natl. Acad. Sci. USA 37, 303-07 (1951) CrossRef
2. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Hafner, New York (1965). Translated by N. Kemmer
3. Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119-178 (1999). arXiv:math/9810105 [math.CO] CrossRef
4. Berele, A., Tenner, B.: Doubly symmetric functions (2009). arXiv:0903.5306 [math.CO]
5. Borodin, A.: Multiplicative central measures on the Schur graph. J. Math. Sci. (N.Y.) 96(5), 3472-477 (1999). In Russian: Zap. Nau?. Semin. POMI 240, 44-2, 290-91 (1997) CrossRef
6. Borodin, A.: Harmonic analysis on the infinite symmetric group, and the Whittaker kernel. St. Petersburg Math. J. 12(5), 733-59 (2001)
7. Borodin, A., Okounkov, A., Olshanski, G.: Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc. 13(3), 481-15 (2000). arXiv:math/9905032 [math.CO] CrossRef
8. Borodin, A., Olshanski, G.: Point processes and the infinite symmetric group. Math. Res. Lett. 5, 799-16 (1998). arXiv:math/9810015 [math.RT] CrossRef
9. Borodin, A., Olshanski, G.: Distributions on partitions, point processes, and the hypergeometric kernel. Commun. Math. Phys. 211(2), 335-58 (2000). arXiv:math/9904010 [math.RT] CrossRef
10. Borodin, A., Olshanski, G.: Harmonic functions on multiplicative graphs and interpolation polynomials. Electron. J. Comb. 7, R28 (2000). arXiv:math/9912124 [math.CO]
11. Borodin, A., Olshanski, G.: Z-Measures on partitions, Robinson–Schensted–Knuth correspondence, and / β=2 random matrix ensembles. In: Bleher, P.M., Its, R.A. (eds.) Random Matrix Models and Their Applications. Math. Sci. Res. Inst. Publ., vol. 40, pp. 71-4. Cambridge University Press, Cambridge (2001). arXiv:math/9905189 [math.CO]
12. Borodin, A., Olshanski, G.: Random partitions and the Gamma kernel. Adv. Math. 194(1), 141-02 (2005). arXiv:math-ph/0305043 CrossRef
13. Borodin, A., Olshanski, G.: Z-measures on partitions and their scaling limits. Eur. J. Comb. 26(6), 795-34 (2005). arXiv:math-ph/0210048 CrossRef
14. Borodin, A., Olshanski, G.: Markov processes on partitions. Probab. Theory Relat. Fields 135(1), 84-52 (2006). arXiv:math-ph/0409075 CrossRef
15. Borodin, A., Olshanski, G.: Meixner polynomials and random partitions. Mosc. Math. J. 6(4), 629-55 (2006). arXiv:math/0609806 [math.PR]
16. Borodin, A., Olshanski, G.: Stochastic dynamics related to Plancherel measure on partitions. In: Kaimanovich, V., Lodkin, A. (eds.) Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, 2, vol. 217, pp. 9-2. AMS, Providence (2006). arXiv:math-ph/0402064
17. Borodin, A., Olshanski, G.: Infinite-dimensional diffusions as limits of random walks on partitions. Probab. Theory Relat. Fields 144(1), 281-18 (2009). arXiv:0706.1034 [math.PR] CrossRef
18. Borodin, A., Olshanski, G.: The Young bouquet and its boundary (2011). arXiv:1110.4458 [math.RT]
19. Boyer, R.: Infinite traces of AF-algebras and characters of / U(?. J. Oper. Theory 9, 205-36 (1983)
20. Dawson, D.: Measure-valued Markov processes. In: école d’été de Probabilités de Saint-Flour XXI-991. Lecture Notes in Math., vol. 1541, pp. 1-60. Springer, Berlin (1993) CrossRef
21. Desrosiers, P., Hallnas, M.: Hermite and Laguerre symmetric functions associated with operators of Calogero-Moser-Sutherland type (2011). arXiv:1103.4593 [math.QA]
22. Edrei, A.: On the generating functions of totally positive sequences. II. J. Anal. Math. 2, 104-09 (1952) CrossRef
23. Eie, B.: The generalized Bessel process corresponding to an Ornstein–Uhlenbeck process. Scand. J. Stat. 10(3), 247-50 (1983)
24. Erdélyi, A. (ed.): Higher Transcendental Functions. McGraw-Hill, New York (1953)
25. Etheridge, A.: An Introduction to Superprocesses. University Lecture Series, vol. 20. Am. Math. Soc., Providence (2000)
26. Etheridge, A., March, P.: A note on superprocesses. Probab. Theory Relat. Fields 89(2), 141-47 (1991) CrossRef
27. Ethier, S., Kurtz, T.: The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Probab. 13(3), 429-52 (1981) CrossRef
28. Ethier, S., Kurtz, T.: Markov Processes: Characterization and Convergence. Wiley-Interscience, New York (1986) CrossRef
29. Ewens, W.J.: Mathematical Population Genetics. Springer, Berlin (1979)
30. Feng, S.: The Poisson–Dirichlet distributions and related topics: models and asymptotic behaviours. Springer, Berlin (2010) CrossRef
31. Fomin, S.: Two-dimensional growth in Dedekind lattices. Master’s thesis, Leningrad State University (1979)
32. Fomin, S.: Duality of graded graphs. J. Algebr. Comb. 3(4), 357-04 (1994) CrossRef
33. Fomin, S., Stanton, D.: Rim hook lattices. St. Petersburg Math. J. 9(5), 1007-016 (1998). Translated from Algebra Anal. 9(5), 140-50 (1997)
34. Fulman, J.: Stein’s method and Plancherel measure of the symmetric group. Trans. Am. Math. Soc. 357(2), 555-70 (2005). arXiv:math/0305423 [math.RT] CrossRef
35. Fulman, J.: Commutation relations and Markov chains. Probab. Theory Relat. Fields 144(1), 99-36 (2009). arXiv:0712.1375 [math.PR] CrossRef
36. Fulman, J.: Mixing time for a random walk on rooted trees. Electron. J. Comb. 16, R139 (2009). arXiv:0908.1141 [math.CO]
37. Hoffman, M.: Combinatorics of rooted trees and Hopf algebras. Trans. Am. Math. Soc. 355(9), 3795-811 (2003) (electronic). arXiv:math/0201253 [math.CO] CrossRef
38. Ivanov, V.: The dimension of skew shifted Young diagrams, and projective characters of the infinite symmetric group. J. Math. Sci. (N.Y.) 96(5), 3517-530 (1999). In Russian: Zap. Nau?. Semin. POMI 240, 115-35 (1997). arXiv:math/0303169 [math.CO] CrossRef
39. Jirina, M.: Branching processes with measure-valued states. In: Trans. Third Prague Conf. on Inf. Th, pp. 333-57 (1964)
40. Karlin, S., McGregor, J.: The classification of birth and death processes. Trans. Am. Math. Soc. 86, 366-00 (1957) CrossRef
41. Karlin, S., McGregor, J.: Linear growth, birth and death processes. J. Math. Mech. 7, 643-62 (1958)
42. Kerov, S.: Combinatorial examples in the theory of AF-algebras. Zap. Nau?. Semin. POMI 172, 55-7 (1989). English translation: J. Sov. Math. 59, 1063-071 (1992)
43. Kerov, S.: Anisotropic Young diagrams and Jack symmetric functions. Funct. Anal. Appl. 34(1), 41-1 (2000). arXiv:math/9712267 [math.CO] CrossRef
44. Kerov, S.: Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis vol. 219. AMS, Providence (2003). Translations of Mathematical Monographs
45. Kerov, S., Okounkov, A., Olshanski, G.: The boundary of Young graph with Jack edge multiplicities. Int. Math. Res. Not. 4, 173-99 (1998). arXiv:q-alg/9703037 CrossRef
46. Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group. A deformation of the regular representation. C.R. Acad. Sci. Paris Ser. I 316, 773-78 (1993)
47. Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group. Invent. Math. 158(3), 551-42 (2004). arXiv:math/0312270 [math.RT] CrossRef
48. Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Tech. Rep., Delft University of Technology and Free University of Amsterdam (1996)
49. K?nig, W.: Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385-47 (2005). arXiv:math/0403090 [math.PR] CrossRef
50. Koornwinder, T.: Krawtchouk polynomials, a unification of two different group theoretic interpretations. SIAM J. Math. Anal. 13(6), 1011-023 (1982) CrossRef
51. Kuperberg, G.: Self-complementary plane partitions by Proctor’s minuscule method. Eur. J. Comb. 15(6), 545-53 (1994). arXiv:math/9411239 [math.CO] CrossRef
52. Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995)
53. Matsumoto, S.: Correlation functions of the shifted Schur measure. J. Math. Soc. Jpn. 57(3), 619-37 (2005). arXiv:math/0312373 [math.CO] CrossRef
54. Matsumoto, S.: Jack deformations of Plancherel measures and traceless Gaussian random matrices. Electron. J. Comb. 15(R149), 1 (2008). arXiv:0810.5619 [math.CO]
55. Nelson, E.: Analytic vectors. Ann. Math. 2(70), 572-15 (1959) CrossRef
56. Okounkov, A.: Quantum immanants and higher Capelli identities. Transform. Groups 1(1-), 99-26 (1996). arXiv:q-alg/9602028 CrossRef
57. Okounkov, A.: Random matrices and random permutations. Int. Math. Res. Not. 2000(20), 1043-095 (2000). arXiv:math/9903176 [math.CO] CrossRef
58. Okounkov, A.: Infinite wedge and random partitions. Sel. Math. New Ser. 7(1), 57-1 (2001). arXiv:math/9907127 [math.RT] CrossRef
59. Okounkov, A.: SL(2) and z-measures. In: Bleher, P.M., Its, A.R. (eds.) Random Matrix Models and Their Applications. Mathematical Sciences Research Institute Publications, vol. 40, pp. 407-20. Cambridge University Press, Cambridge (2001). arXiv:math/0002135 [math.RT]
60. Okounkov, A.: Symmetric functions and random partitions. In: Fomin, S. (ed.) Symmetric Functions 2001: Surveys of Developments and Perspectives. Kluwer Academic, Norwell (2002). arXiv:math/0309074 [math.CO]
61. Okounkov, A., Olshanski, G.: Shifted Jack polynomials, binomial formula, and applications. Math. Res. Lett. 4(1), 69-8 (1997). arXiv:q-alg/9608020 CrossRef
62. Okounkov, A., Olshanski, G.: Shifted Schur functions. Algebra Anal. 9(2), 73-46 (1997). Translation in St. Petersburg Math. J. 9(2), 239-00 (1998). arXiv:q-alg/9605042
63. Okounkov, A., Olshanski, G.: Asymptotics of Jack polynomials as the number of variables goes to infinity. Int. Math. Res. Not. 1998(13), 641-82 (1998). arXiv:q-alg/9709011 CrossRef
64. Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16(3), 581-03 (2003). arXiv:math/0107056 [math.CO] CrossRef
65. Olshanski, G.: Fock space and time-dependent determinantal point processes (2008). Unpublished work
66. Olshanski, G.: Anisotropic Young diagrams and infinite-dimensional diffusion processes with the Jack parameter. Int. Math. Res. Not. 2010(6), 1102-166 (2010). arXiv:0902.3395 [math.PR]
67. Olshanski, G.: Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes. Zap. Nau?. Semin. POMI 378, 81-10, 230 (2010). arXiv:1009.2037 [math.CO]
68. Olshanski, G.: Laguerre and Meixner orthogonal bases in the algebra of symmetric functions (2011). arXiv:1103.5848 [math.CO]
69. Olshanski, G., Regev, A., Vershik, A.: Frobenius–Schur functions. In: Joseph, A., Melnikov, A., Rentschler, R. (eds.) Studies in Memory of Issai Schur. Progress in Mathematics, vol. 210, pp. 251-00. Birkhauser, Basel (2003). arXiv:math/0110077 [math.CO] CrossRef
70. Petrov, L.: A two-parameter family of infinite-dimensional diffusions in the Kingman simplex. Funct. Anal. Appl. 43(4), 279-96 (2009). arXiv:0708.1930 [math.PR] CrossRef
71. Petrov, L.: Random strict partitions and determinantal point processes. Electron. Commun. Probab. 15, 162-75 (2010). arXiv:1002.2714 [math.PR] CrossRef
72. Petrov, L.: Random walks on strict partitions. J. Math. Sci. 168(3), 437-63 (2010). In Russian: Zap. Nau?. Semin. POMI 373, 226-72 (2009). arXiv:0904.1823 [math.PR] CrossRef
73. Petrov, L.: Pfaffian stochastic dynamics of strict partitions. Electron. J. Probab. 16, 2246-295 (2011). arXiv:1011.3329 [math.PR] CrossRef
74. Pitman, J.: The two-parameter generalization of Ewens-random partition structure. Technical report 345, Dept. Statistics, U.C. Berkeley, (1992). http://www.stat.berkeley.edu/tech-reports/
75. Pitman, J., Yor, M.: Two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25(2), 855-00 (1997) CrossRef
76. Pr?hofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071-106 (2002). arXiv:math.PR/0105240 CrossRef
77. Proctor, R.: Solution of two difficult combinatorial problems with linear algebra. Am. Math. Mon. 89(10), 721-34 (1982) CrossRef
78. Proctor, R.: Bruhat lattices, plane partition generating functions, and minuscule representations. Eur. J. Comb. 5(4), 331-50 (1984) CrossRef
79. Pukanszky, L.: The Plancherel formula for the universal covering group of / SL(2,?. Math. Ann. 156(2), 96-43 (1964) CrossRef
80. Riesz, M.: Sur le probleme des moments et le théoreme de Parseval correspondant. Acta Litt. Acad. Sci. Szeged 1, 209-25 (1923)
81. Rozhkovskaya, N.: Multiplicative distributions on Young graph. J. Math. Sci. (N.Y.) 96(5), 3600-608 (1999). In Russian: Zap. Nau?. Semin. POMI 240, 245-56 (1997) CrossRef
82. Ruggiero, M., Walker, S.: Countable representation for infinite dimensional diffusions derived from the two-parameter Poisson–Dirichlet process. Electron. Commun. Probab. 14, 501-17 (2009) CrossRef
83. Stanley, R.: Differential posets. J. Am. Math. Soc. 1(4), 919-61 (1988) CrossRef
84. Stanley, R.: Variations on differential posets. In: Invariant Theory and Tableaux, Minneapolis, MN, 1988. IMA Vol. Math. Appl., vol. 19, pp. 145-65. Springer, New York (1990)
85. Stanley, R.: Enumerative Combinatorics. Vol. 1. Cambridge University Press, Cambridge (1997). With a foreword by Gian-Carlo Rota, corrected reprint of the 1986 original CrossRef
86. Stanley, R.: Enumerative Combinatorics. Vol. 2. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin CrossRef
87. Stembridge, J.: A characterization of supersymmetric polynomials. J. Algebra 95, 439-44 (1985) CrossRef
88. Stembridge, J.: On minuscule representations, plane partitions and involutions in complex Lie groups. Duke Math. J. 73(2), 469-90 (1994) CrossRef
89. Strahov, E.: The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel. Adv. Math. 224(1), 130-68 (2010). arXiv:0905.1994 [math-ph] CrossRef
90. Strahov, E.: Z-measures on partitions related to the infinite Gelfand pair ( / S(2?, / H(?). J. Algebra 323(2), 349-70 (2010). arXiv:0904.1719 [math.RT] CrossRef
91. Thoma, E.: Die unzerlegbaren, positive-definiten Klassenfunktionen der abz?hlbar unendlichen, symmetrischen Gruppe. Math. Z. 85, 40-1 (1964) CrossRef
92. Tracy, C., Widom, H.: A Limit Theorem for Shifted Schur Measures. Duke Math. J. 123, 171-08 (2004). arXiv:math/0210255 [math.PR] CrossRef
93. Vershik, A.: Statistical mechanics of combinatorial partitions, and their limit shapes. Funct. Anal. Appl. 30, 90-05 (1996) CrossRef
94. Vershik, A., Kerov, S.: Asymptotic theory of the characters of a symmetric group. Funkc. Anal. Prilozh. 15(4), 15-7, 96 (1981)
95. Vershik, A., Kerov, S.: Characters and factor representations of the infinite symmetric group. Dokl. Akad. Nauk SSSR 257(5), 1037-040 (1981)
96. Vershik, A., Kerov, S.: Characters and factor-representations of the infinite unitary group. Dokl. Akad. Nauk SSSR 267(2), 272-76 (1982)
97. Vershik, A., Kerov, S.: Characters, factor representations and / K-functor of the infinite symmetric group. In: Operator algebras and group representations, Vol. II. Neptun 1980. Monogr. Stud. Math., vol. 18, pp. 23-2. Pitman, Boston (1984)
98. Vershik, A., Kerov, S.: The Grothendieck group of the infinite symmetric group and symmetric functions with the elements of the / K 0-functor theory of AF-algebras. In: Adv. Stud. Contemp. Math., vol. 7, pp. 36-14. Gordon & Breach, New York (1990)
99. Vershik, A., Nikitin, P.: Description of the characters and factor representations of the infinite symmetric inverse semigroup. Funct. Anal. Appl. 45(1), 13-4 (2011). arXiv:1102.4425 [math.RT] CrossRef
100. Vilenkin, N., Klimyk, A.: Representations of the group / SU(1,1), and the Krawtchouk–Meixner functions. Dokl. Akad. Nauk Ukrain. SSR Ser. A 6, 12-6 (1988)
101. Vilenkin, N., Klimyk, A.: Representations of Lie groups and special functions. In: Kirillov, A. (ed.) Representation Theory and Noncommutative Harmonic Analysis II, pp. 137-59. Springer, Berlin (1995). Translation of VINITI, vol.?59 (1990)
102. Voiculescu, D.: Representations factorielles de type / II 1 de / U(?. J. Math. Pures Appl. 55, 1-0 (1976) - 作者单位：Leonid Petrov (1) (2)
1. Department of Mathematics, Northeastern University, 360 Huntington ave., Boston, MA, 02115, USA
2. Dobrushin Mathematics Laboratory, Kharkevich Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia - ISSN：1572-9192
文摘
We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes Kerov’s operators which first appeared in Okounkov (Random Matrix Models and Their Applications, pp.?407-20, Cambridge University Press, Cambridge, 2001), and also stems from the study of duality of graded graphs in Fomin (J. Algebr. Comb., 3(4):357-04, 1994). Our main object is a countable branching graph carrying an $\mathfrak {sl}(2,\mathbb{C})$ -module of a special kind. Using this structure, we introduce distinguished probability measures on the floors of the graph, and define two related types of Markov dynamics associated with these measures. We study spectral properties of the dynamics, and our main result is the explicit description of eigenfunctions of the Markov generator of one of the processes. For the Young graph our approach reconstructs the z-measures on partitions and the associated dynamics studied by Borodin and Olshanski (Probab. Theory Relat. Fields, 135(1):84-52, 2006; Probab. Theory Relat. Fields, 144(1):281-18, 2009). The generator of the jump dynamics on the Young graph corresponding to the z-measures is diagonal in the basis of the Meixner symmetric functions introduced recently by Olshanski (Zap. Nau?. Semin. POMI, 378:81-00, 230, 2010; Laguerre and Meixner orthogonal bases in the algebra of symmetric functions, 2011). We give new proofs to some of the results of these two papers. Other graphs to which our technique is applicable include the Pascal triangle, the Kingman graph (with the two-parameter Poisson–Dirichlet measures), the Schur graph and the general Young graph with Jack edge multiplicities.