Abstract
Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinite-dimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the so-called z-measures, appeared earlier in the problem of harmonic analysis for the infinite symmetric group. The generators of the processes are explicitly described.
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Borodin, A.: Harmonic analysis on the infinite symmetric group and the Whittaker kernel. St. Petersb. Math. J. 12(5), 733–759 (2001)
Borodin, A., Olshanski, G.: Point processes and the infinite symmetric group. Math. Res. Lett. 5, 799–816 (1998). arXiv: math.RT/9810015
Borodin, A., Olshanski, G.: Distributions on partitions, point processes and the hypergeometric kernel. Comm. Math. Phys. 211, 335–358 (2000). arXiv: math. RT/9904010
Borodin, A., Olshanski, G.: Harmonic functions on multiplicative graphs and interpolation polynomials. Electron. J. Comb. 7 (2000), paper #R28. arXiv: math/9912124
Borodin, A., Olshanski, G.: Z-Measures on partitions, Robinson–Schensted–Knuth correspondence, and β = 2 random matrix ensembles. In: Bleher, P.M., Its, A.R. (eds.) Random Matrix Models and their Applications, vol. 40, pp. 71–94. Mathematical Sciences Research Institute Publications, Cambridge University Press, London (2001). arXiv: math/9905189
Borodin, A., Olshanski, G.: Random partitions and the Gamma kernel. Adv. Math. 194(1), 141–202 (2005). arXiv: math-ph/0305043
Borodin, A., Olshanski, G.: Markov processes on partitions. Prob. Theory Relat. Fields 135(1), 84–152 (2006). arXiv: math-ph/0409075
Borodin, A., Olshanski, G.: Meixner polynomials and random partitions. Moscow Math. J. 6(4), 629–655 (2006). arXiv: math.PR/0609806
Edrei, A.: On the generating functions of totally positive sequences II. J. Anal. Math. 2, 104–109 (1952)
Ethier, S.N., Kurtz, T.G.: The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452 (1981)
Ethier, S.N., Kurtz, T.G.: Markov Processes—Characterization and Convergence. Wiley-Interscience, New York (1986)
Frenkel, I.B., Wang, W.: Virasoro algebra and wreath product convolution. J. Algebra 242(2), 656–671 (2001)
Fulman, J.: Stein’s method and Plancherel measure of the symmetric group. Trans. Am. Math. Soc. 357(2), 555–570 (2005). arXiv: math.RT/0305423
Ivanov, V., Olshanski, G.: Kerov’s central limit theorem for the Plancherel measure on Young diagrams. In: Fomin, S. (ed.) Symmetric functions 2001. Surveys of developments and perspectives. Proc. NATO Advanced Study Institute, pp. 93–151. Kluwer, Dordrecht (2002). arXiv: math/0304010
Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov chains. Springer, New York (1976)
Kerov, S.V.: Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, 201 pp. American Mathematical Society, Providence (2003)
Kerov, S., Okounkov, A., Olshanski, G.: The boundary of Young graph with Jack edge multiplicities. Intern. Math. Res. Notices (4), 173–199 (1998). arXiv: q-alg/9703037
Kerov, S., Olshanski, G.: Polynomial functions on the set of Young diagrams. Comptes Rendus Acad. Sci. Paris Sér. I 319, 121–126 (1994)
Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group. A deformation of the regular representation. Comptes Rendus Acad. Sci. Paris, Sér. I 316, 773–778 (1993)
Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group. Invent. Math. 158, 551–642 (2004). arXiv: math.RT/0312270
Lamperti, J.: Stochastic Processes, A Survey of the Mathematical Theory. Springer, New York (1977)
Lascoux, A., Thibon, J.-Y.: Vertex operators and the class algebras of symmetric groups. J. Math. Sci. (NY) 121(3), 2380–2392 (2004). arXiv: math/0102041
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)
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. 407–420. Cambridge University Press, London (2001) arXiv:math.RT/0002136
Okounkov, A., Olshanski, G.: Shifted Schur functions. Algebra Anal. 9(2) 73–146 (1997) (Russian); English translation: St. Petersb. Math. J. 9(2), 239–300 (1998). arXiv:q-alg/9605042
Olshanski, G.: Point processes related to the infinite symmetric group. In: Duval, Ch., et al. (eds.) The orbit method in geometry and physics: in honor of A. A. Kirillov. Progress in Mathematics, vol. 213, pp. 349–393. Birkhäuser (2003). arXiv: math.RT/9804086
Olshanski, G.: An introduction to harmonic analysis on the infinite symmetric group. In: Vershik, A. M. (ed.) Asymptotic combinatorics with applications to mathematical physics. A European Mathematical Summer School 9–20 July 2001. Springer Lecture Notes on Mathematics, vol. 1815, pp. 127–160. Euler Institute, St. Petersburg (2003). arXiv: math.RT/0311369
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. 251–300. Birkhäuser (2003). arXiv: math/0110077
Overbeck, L., Röckner, M., Schmuland, B.: An analytic approach to Fleming–Viot processes with interactive selection. Ann. Prob. 23, 1–36 (1995)
Petrov, L.A.: Two-parameter family of diffusion processes in the Kingman simplex (2007, preprint). arXiv:0708.1930 [math.PR]
Schmuland, B.: A result on the infinitely many neutral alleles diffusion model. J. Appl. Prob. 28, 253–267 (1991)
Thoma, E.: Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Zeitschr. 85, 40–61 (1964)
Vershik, A.M., Kerov, S.V.: Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl. 15, 246–255 (1981)
Wentzell, A.D.: A Course in the Theory of Stochastic Processes. McGraw-Hill International, New York (1981)
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Borodin, A., Olshanski, G. Infinite-dimensional diffusions as limits of random walks on partitions. Probab. Theory Relat. Fields 144, 281–318 (2009). https://doi.org/10.1007/s00440-008-0148-8
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DOI: https://doi.org/10.1007/s00440-008-0148-8