Skip to main content
SpringerLink
Log in

Asymptotic theory of path spaces of graded graphs and its applications

  • Takagi Lectures
  • Published:
Japanese Journal of Mathematics Aims and scope

Abstract

The survey covers several topics related to the asymptotic structure of various combinatorial and analytic objects such as the path spaces in graded graphs (Bratteli diagrams), invariant measures with respect to countable groups, etc. The main subject is the asymptotic structure of filtrations and a new notion of standardness. All graded graphs and all filtrations of Borel or measure spaces can be divided into two classes: the standard ones, which have a regular behavior at infinity, and the other ones. Depending on this property, the list of invariant measures can either be well parameterized or have no good parametrization at all. One of the main results is a general standardness criterion for filtrations. We consider some old and new examples which illustrate the usefulness of this point of view and the breadth of its applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abert M., Bergeron N., Biringer I., Gelander T., Nikolov N., Raimbault J., Samet I: On the growth of Betti numbers of locally symmetric spaces. C. R. Math. Acad. Sci. Paris, 349, 831–835 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. D.J. Aldous, Exchangeability and related topics, In: École d’Été de Probabilités de Saint-Flour, XIII–1983, Lecture Notes in Math., 1117, Springer-Verlag, 1985, pp. 1–198.

  3. Bratteli O.: Inductive limits of finite dimensional C*-algebras. Trans. Amer. Math. Soc., 171, 195–234 (1972)

    MathSciNet  MATH  Google Scholar 

  4. O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2, Texts Monogr. Phys., Springer-Verlag, 1997.

  5. Bufetov A.I.: On the Vershik–Kerov conjecture concerning the Shannon–McMillan–Breiman theorem for the Plancherel family of measures on the space of Young diagrams. Geom. Funct. Anal., 22, 938–975 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. L.A. Bunimovich, S.G. Dani, R.L. Dobrushin, M.V. Jakobson, I.P. Kornfeld, N.B. Maslova, Ya.B. Pesin, Ya.G. Sinai, J. Smillie, Yu.M. Sukhov and A.M. Vershik, Dynamical Systems, Ergodic Theory and Applications, Encyclopaedia Math. Sci., 100, Springer-Verlag, 2000.

  7. T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli, Representation Theory of the Symmetric Groups. The Okounkov–Vershik Approach, Character Formulas, and Partition Algebras, Cambridge Stud. Adv. Math., 121, Cambridge Univ. Press, Cambridge, 2010.

  8. Connes A., Feldman J., Weiss B.: An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynamical Systems, 1, 431–450 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. Effros E.G., Handelman D.E., Shen C.L.: Dimension groups and their affine representations. Amer. J. Math., 102, 385–402 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Elliott G.A.: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra, 38, 29–44 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Giordano T., Putnam I.F., Skau C.: Full groups of Cantor minimal systems. Israel J. Math., 111, 285–320 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gnedin A., Olshanski G.: q-exchangeability via quasi-invariance. Ann. Probab., 38, 2103–2135 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Goodman F.M., Kerov S.V.: The Martin boundary of the Young–Fibonaci lattice. J. Algebraic Combin., 11, 17–48 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Greschonig G., Schmidt K.: Ergodic decomposition of quasi-invariant probability measures. Colloq. Math., 85, 495–514 (2000)

    MathSciNet  MATH  Google Scholar 

  15. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math., 152, Birkhäuser Boston, Boston, MA, 1999.

  16. Hajian A., Ito Y., Kakutani S.: Invariant measures and orbits of dissipative transformations. Advances in Math., 9, 52–65 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ito S.: A construction of transversal flows for maximal Markov automorphisms. Tokyo J. Math., 1, 305–324 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Janvresse É., de la Rue T.: The Pascal adic transformation is loosely Bernoulli. Ann. Inst. H. Poincaré Probab. Statist., 40, 133–139 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Janvresse É., de la Rue T., Velenik Y.: Self-similar corrections to the ergodic theorem for the Pascal-adic transformation. Stoch. Dyn., 5, 1–25 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Kakutani, A problem of equidistribution on the unit interval [0, 1], In: Meadure Theory, Lecture Notes in Math., 541, Springer-Verlag, 1976, pp. 369–375.

  21. O. Kallenberg, Probabilistic Symmetries and Invariance Principles, Springer-Verlag, 2005.

  22. L.V. Kantorovich, On the translocation of masses, C. R. (Doklady) Akad. Sci. URSS (N.S.), 37 (1942), 199–201. English translation: J. Math. Sci., 133 (2006), 1381–1382.

  23. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995.

  24. A.S. Kechris, The structure of Borel equivalence relations in Polish spaces, In: Set Theory of the Continuum, (eds. H. Judah, W. Just and W.H. Woodin), Math. Sci. Res. Inst. Publ., 26, Springer-Verlag, 1992, pp. 89–102.

  25. S.V. Kerov, Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, Transl. Math. Monogr., 219, Amer. Math. Soc., Providence, RI, 2003.

  26. S. Kerov, A. Okounkov and G. Olshanski, The boundary of the Young graph with Jack edge multiplicities, Internat. Math. Res. Notices, 1998, no. 4, 173–199.

  27. Liebermann A.: The structure of certain unitary representations of infinite symmetric groups. Trans. Amer. Math. Soc., 164, 189–198 (1972)

    Article  MathSciNet  Google Scholar 

  28. A.A. Lodkin and A.R. Minabutdinov, Limiting curves for the Pascal adic transformation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 437 (2015), 145–183. English translation to appear in J. Math. Sci. (2016).

  29. Méla X., Petersen K.: Dynamical properties of the Pascal adic transformation. Ergodic Theory Dynam. Systems, 25, 227–256 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Okounkov A.: On representations of the infinite symmetric group. J. Math. Sci. (New York), 96, 3550–3589 (1999)

    Article  MathSciNet  Google Scholar 

  31. Okounkov A., Vershik A.M.: A new approach to representation theory of symmetric groups. Selecta Math. (N.S.), 2, 581–605 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ol’shanskiĭ G.I.: Unitary representations of (G,K)-pairs that are connected with the infinite symmetric group \({S(\infty)}\). Leningrad Math. J., 1, 983–1014 (1990)

    MathSciNet  Google Scholar 

  33. R.R. Phelps, Lectures on Choquet’s Theorem. 2nd ed., Lecture Notes in Math., 1757, Springer-Verlag, 2001.

  34. Pimsner M.V.: Embedding some transformation group C*-algebra into AF-algebras. Ergodic Theory Dynam. Systems, 3, 613–626 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Rørdam and E. Størmer, Classification of Nuclear C *-Algebras, Encyclopaedia Math. Sci., 126, Springer-Verlag, 2002.

  36. Schmidt K.: A probabilistic proof of ergodic decomposition. Sankhyā Ser. A, 40, 10–18 (1978)

    MathSciNet  MATH  Google Scholar 

  37. Ş. Strătilă and D.-V. Voiculescu, Representations of AF-Algebras and of the Group \({U_{\infty}}\), Lecture Notes in Math., 486, Springer-Verlag, 1975.

  38. Thoma E.: Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen. symmetrischen Gruppe. Math. Z., 85, 40–61 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  39. Thoma E.: Eine Charakterisierung diskreter Gruppen vom Typ I. Invent. Math., 6, 190–196 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  40. A.M. Vershik, A theorem on the lacunary isomorphism of monotonic sequences of partitionings, Funktcional. Anal. i Prilož, 2 (1968), no. 3, 17–21.

  41. Vershik A.M.: Decreasing sequences of measurable partitions and their applications. Sov. Math. Dokl., 11, 1007–1011 (1970)

    MATH  Google Scholar 

  42. A.M. Vershik, Approximation in measure theory, Dissertation, Leningrad State Univ., Leningrad, 1973.

  43. Vershik A.M.: Four definitions of the scale of an automorphism. Functional Anal. Appl., 7, 169–181 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  44. Vershik A.M.: Uniform algebraic approximation of shift and multiplication operators. Dokl. Akad. Nauk SSSR, 259, 526–529 (1981)

    MathSciNet  MATH  Google Scholar 

  45. Vershik A.M.: A theorem on the Markov periodic approximation in ergodic theory. J. Sov. Math., 28, 667–674 (1985)

    Article  MATH  Google Scholar 

  46. Vershik A.M.: Theory of decreasing sequences of measurable partitions. St. Petersburg Math. J., 6, 705–761 (1995)

    MathSciNet  Google Scholar 

  47. A.M. Vershik, Asymptotic combinatorics and algebraic analysis, In: Proc. of the International Congress of Mathematicians, Zürich, 1994, Vol. II, Birkhäuser, Basel, 1995, pp. 1384–1394.

  48. Vershik A.M.: Statistical mechanics of combinatorial partitions, and their limit configurations. Funct. Anal. Appl., 30, 90–105 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  49. Vershik A.M.: The universal Uryson space, Gromov’s metric triples, and random metrics on the series of natural numbers. Russian Math. Surveys, 53, 921–928 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  50. Vershik A.M.: Classification of measurable functions of several arguments, and invariantly distributed random matrices. Funct. Anal. Appl., 36, 93–105 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. A.M. Vershik, Random and universal metric spaces, In: Fundamental Mathematics Today, (eds. S.K. Lando and O.K. Sheinman), Nezavis. Mosk. Univ., Moscow, 2003, pp. 54–88.

  52. Vershik A.M.: Random metric spaces and universality. Russian Math. Surveys, 59, 259–295 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  53. Vershik A.M.: The Pascal automorphism has a continuous spectrum. Funct. Anal. Appl., 45, 173–186 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Vershik A.M.: Nonfree actions of countable groups and their characters. J. Math. Sci. (N. Y.), 174, 1–6 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Vershik A.M.: Totally nonfree actions and the infinite symmetric group. Mosc. Math. J., 12, 193–212 (2012)

    MathSciNet  MATH  Google Scholar 

  56. Vershik A.M.: On the classification of measurable functions of several variables. J. Math. Sci. (N. Y.), 190, 427–437 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Vershik A.M.: Long history of the Monge–Kantorovich transportation problem. Math. Intelligencer, 35, 1–9 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  58. Vershik A.M.: Intrinsic metric on graded graphs, standardness, and invariant measures. J. Math. Sci. (N. Y.), 200, 677–681 (2014)

    Article  MATH  Google Scholar 

  59. Vershik A.M.: The problem of describing central measures on the path spaces of graded graphs. Funct. Anal. Appl., 48, 256–271 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  60. Vershik A.M.: Several remarks on Pascal automorphism and infinite ergodic theory. Armen. J. Math., 7, 85–96 (2015)

    MathSciNet  MATH  Google Scholar 

  61. A.M. Vershik, Smoothness and standardness in the theory of AF-algebras and in the problem on invariant measures, In: Probability and Statistics in St. Petersburg, Proc. Sympos. Pure Math., 91, Amer. Math. Soc., Providence, RI, 2015. Preliminary version: preprint, arXiv:1304.2193.

  62. A.M. Vershik, Invariant measures: new aspects of dynamics, combinatorics and representation theory, In: The Fifteenth Takagi Lectures, Math. Soc. Japan, Tokyo, 2015, pp. 39–79.

  63. Vershik A.M.: Equipped graded graphs, projective limits of simplices, and their boundaries. J. Math. Sci. (N. Y.), 209, 860–873 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  64. Vershik A.M.: Standardness as an invariant formulation of independence. Funct. Anal. Appl., 49, 253–263 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  65. Vershik A.M., Gorbul’skiĭ A.D.: Scaled entropy of filtrations of \({\sigma}\) -algebras. Teor. Veroyatn. Primen., 52, 446–467 (2007)

    Article  MathSciNet  Google Scholar 

  66. A.M. Vershik and U. Haböck, On the classification problem of measurable functions in several variables and on matrix distributions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), 119–143. English translation to appear in J. Math. Sci. (2016); preprint, arXiv:1512.06760.

  67. Vershik A.M., Kerov S.V.: Characters and factor representations of the infinite symmetric group. Soviet Math. Dokl., 23, 389–392 (1981)

    MathSciNet  MATH  Google Scholar 

  68. Vershik A.M., Kerov S.V.: Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl., 15, 246–255 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  69. Vershik A.M., Kerov S.V.: Asymptotics of the largest and the typical dimensions of irreducible representations of a symmetric group. Funct. Anal. Appl., 19, 21–31 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  70. A.M. Vershik and S.V. Kerov, Locally semisimple algebras. Combinatorial theory and the K 0-functor, J. Soviet Math., 38 (1987), 1701–1733.

  71. VershikA.M. Malyutin A.V.: Phase transition in the exit boundary problem for random walks on groups. Funct. Anal. Appl., 49, 86–96 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  72. Vershik A.M., Nikitin P.P.: Description of characters and factor representations of the infinite symmetric inverse semigroup. Funct. Anal. Appl., 45, 13–24 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  73. Vershik A.M., Zatitskiĭ P.B., Petrov F.V.: Geometry and dynamics of admissible metrics in measure spaces. Cent. Eur. J. Math., 11, 379–400 (2013)

    MathSciNet  MATH  Google Scholar 

  74. Vershik A.M., Zatitskiĭ P.B., Petrov F.V.: Virtual continuity of measurable functions of several variables and its applications. Russian Math. Surveys, 69, 1031–1063 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  75. G. Winkler, Choquet Order and Simplices with Application in Probabilistic Models, Lecture Notes in Math., 1145, Springer-Verlag, 1985.

Download references

Author information

Authors and Affiliations

  1. St. Petersburg Department of Steklov Institute of Mathematics, Mathematical Department of St. Petersburg State University, Moscow Institute for Information Transmission Problems, 27 Fontanka, St. Petersburg, 191023, Russia

    Anatoly M. Vershik

Authors
  1. Anatoly M. Vershik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatoly M. Vershik.

Additional information

Communicated by: Toshiyuki Kobayashi

This article is based on the 15th Takagi Lectures that the author delivered at Tohoku University on June 27 and 28, 2015

Supported by the Russian Science Foundation grant 14-11-00581.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vershik, A.M. Asymptotic theory of path spaces of graded graphs and its applications. Jpn. J. Math. 11, 151–218 (2016). https://doi.org/10.1007/s11537-016-1527-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-016-1527-z

Keywords and phrases

Mathematics Subject Classification (2010)

Search

Navigation