Japanese Journal of Mathematics

, Volume 11, Issue 2, pp 151–218 | Cite as

Asymptotic theory of path spaces of graded graphs and its applications

  • Anatoly M. VershikEmail author
Takagi Lectures


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.

Mathematics Subject Classification (2010)

37A55 60B05 52A23 

Keywords and phrases

graded graph Markov compactum cotransition probability central measure filtration standardness limit shape 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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.Google Scholar
  3. 3.
    Bratteli O.: Inductive limits of finite dimensional C*-algebras. Trans. Amer. Math. Soc., 171, 195–234 (1972)MathSciNetzbMATHGoogle Scholar
  4. 4.
    O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2, Texts Monogr. Phys., Springer-Verlag, 1997.Google Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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.Google Scholar
  7. 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.Google Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Effros E.G., Handelman D.E., Shen C.L.: Dimension groups and their affine representations. Amer. J. Math., 102, 385–402 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Elliott G.A.: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra, 38, 29–44 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Giordano T., Putnam I.F., Skau C.: Full groups of Cantor minimal systems. Israel J. Math., 111, 285–320 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gnedin A., Olshanski G.: q-exchangeability via quasi-invariance. Ann. Probab., 38, 2103–2135 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goodman F.M., Kerov S.V.: The Martin boundary of the Young–Fibonaci lattice. J. Algebraic Combin., 11, 17–48 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Greschonig G., Schmidt K.: Ergodic decomposition of quasi-invariant probability measures. Colloq. Math., 85, 495–514 (2000)MathSciNetzbMATHGoogle Scholar
  15. 15.
    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math., 152, Birkhäuser Boston, Boston, MA, 1999.Google Scholar
  16. 16.
    Hajian A., Ito Y., Kakutani S.: Invariant measures and orbits of dissipative transformations. Advances in Math., 9, 52–65 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ito S.: A construction of transversal flows for maximal Markov automorphisms. Tokyo J. Math., 1, 305–324 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Janvresse É., de la Rue T.: The Pascal adic transformation is loosely Bernoulli. Ann. Inst. H. Poincaré Probab. Statist., 40, 133–139 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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.Google Scholar
  21. 21.
    O. Kallenberg, Probabilistic Symmetries and Invariance Principles, Springer-Verlag, 2005.Google Scholar
  22. 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.Google Scholar
  23. 23.
    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995.Google Scholar
  24. 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.Google Scholar
  25. 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.Google Scholar
  26. 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.Google Scholar
  27. 27.
    Liebermann A.: The structure of certain unitary representations of infinite symmetric groups. Trans. Amer. Math. Soc., 164, 189–198 (1972)MathSciNetCrossRefGoogle Scholar
  28. 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).Google Scholar
  29. 29.
    Méla X., Petersen K.: Dynamical properties of the Pascal adic transformation. Ergodic Theory Dynam. Systems, 25, 227–256 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Okounkov A.: On representations of the infinite symmetric group. J. Math. Sci. (New York), 96, 3550–3589 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Okounkov A., Vershik A.M.: A new approach to representation theory of symmetric groups. Selecta Math. (N.S.), 2, 581–605 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)MathSciNetGoogle Scholar
  33. 33.
    R.R. Phelps, Lectures on Choquet’s Theorem. 2nd ed., Lecture Notes in Math., 1757, Springer-Verlag, 2001.Google Scholar
  34. 34.
    Pimsner M.V.: Embedding some transformation group C*-algebra into AF-algebras. Ergodic Theory Dynam. Systems, 3, 613–626 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    M. Rørdam and E. Størmer, Classification of Nuclear C *-Algebras, Encyclopaedia Math. Sci., 126, Springer-Verlag, 2002.Google Scholar
  36. 36.
    Schmidt K.: A probabilistic proof of ergodic decomposition. Sankhyā Ser. A, 40, 10–18 (1978)MathSciNetzbMATHGoogle Scholar
  37. 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.Google Scholar
  38. 38.
    Thoma E.: Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen. symmetrischen Gruppe. Math. Z., 85, 40–61 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Thoma E.: Eine Charakterisierung diskreter Gruppen vom Typ I. Invent. Math., 6, 190–196 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 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.Google Scholar
  41. 41.
    Vershik A.M.: Decreasing sequences of measurable partitions and their applications. Sov. Math. Dokl., 11, 1007–1011 (1970)zbMATHGoogle Scholar
  42. 42.
    A.M. Vershik, Approximation in measure theory, Dissertation, Leningrad State Univ., Leningrad, 1973.Google Scholar
  43. 43.
    Vershik A.M.: Four definitions of the scale of an automorphism. Functional Anal. Appl., 7, 169–181 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Vershik A.M.: Uniform algebraic approximation of shift and multiplication operators. Dokl. Akad. Nauk SSSR, 259, 526–529 (1981)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Vershik A.M.: A theorem on the Markov periodic approximation in ergodic theory. J. Sov. Math., 28, 667–674 (1985)CrossRefzbMATHGoogle Scholar
  46. 46.
    Vershik A.M.: Theory of decreasing sequences of measurable partitions. St. Petersburg Math. J., 6, 705–761 (1995)MathSciNetGoogle Scholar
  47. 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.Google Scholar
  48. 48.
    Vershik A.M.: Statistical mechanics of combinatorial partitions, and their limit configurations. Funct. Anal. Appl., 30, 90–105 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Vershik A.M.: Classification of measurable functions of several arguments, and invariantly distributed random matrices. Funct. Anal. Appl., 36, 93–105 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 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.Google Scholar
  52. 52.
    Vershik A.M.: Random metric spaces and universality. Russian Math. Surveys, 59, 259–295 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Vershik A.M.: The Pascal automorphism has a continuous spectrum. Funct. Anal. Appl., 45, 173–186 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Vershik A.M.: Nonfree actions of countable groups and their characters. J. Math. Sci. (N. Y.), 174, 1–6 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Vershik A.M.: Totally nonfree actions and the infinite symmetric group. Mosc. Math. J., 12, 193–212 (2012)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Vershik A.M.: On the classification of measurable functions of several variables. J. Math. Sci. (N. Y.), 190, 427–437 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Vershik A.M.: Long history of the Monge–Kantorovich transportation problem. Math. Intelligencer, 35, 1–9 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Vershik A.M.: Intrinsic metric on graded graphs, standardness, and invariant measures. J. Math. Sci. (N. Y.), 200, 677–681 (2014)CrossRefzbMATHGoogle Scholar
  59. 59.
    Vershik A.M.: The problem of describing central measures on the path spaces of graded graphs. Funct. Anal. Appl., 48, 256–271 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Vershik A.M.: Several remarks on Pascal automorphism and infinite ergodic theory. Armen. J. Math., 7, 85–96 (2015)MathSciNetzbMATHGoogle Scholar
  61. 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. 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.Google Scholar
  63. 63.
    Vershik A.M.: Equipped graded graphs, projective limits of simplices, and their boundaries. J. Math. Sci. (N. Y.), 209, 860–873 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Vershik A.M.: Standardness as an invariant formulation of independence. Funct. Anal. Appl., 49, 253–263 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Vershik A.M., Gorbul’skiĭ A.D.: Scaled entropy of filtrations of \({\sigma}\) -algebras. Teor. Veroyatn. Primen., 52, 446–467 (2007)MathSciNetCrossRefGoogle Scholar
  66. 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. 67.
    Vershik A.M., Kerov S.V.: Characters and factor representations of the infinite symmetric group. Soviet Math. Dokl., 23, 389–392 (1981)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Vershik A.M., Kerov S.V.: Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl., 15, 246–255 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 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.Google Scholar
  71. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 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)MathSciNetzbMATHGoogle Scholar
  74. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    G. Winkler, Choquet Order and Simplices with Application in Probabilistic Models, Lecture Notes in Math., 1145, Springer-Verlag, 1985.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2016

Authors and Affiliations

  1. 1.St. Petersburg Department of Steklov Institute of MathematicsMathematical Department of St. Petersburg State University, Moscow Institute for Information Transmission ProblemsSt. PetersburgRussia

Personalised recommendations