Annals of Probability

Universality of the limit shape of convex lattice polygonal lines

Leonid V. Bogachev and Sakhavat M. Zarbaliev

Full-text: Open access


Let Πn be the set of convex polygonal lines Γ with vertices on ℤ+2 and fixed endpoints 0 = (0, 0) and n = (n1, n2). We are concerned with the limit shape, as n → ∞, of “typical” ΓΠn with respect to a parametric family of probability measures {Pnr, 0 < r < ∞} on Πn, including the uniform distribution (r = 1) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. Bárány and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class {Pnr}, even though Pnr (r ≠ 1) and Pn1 are asymptotically singular. Measures Pnr are constructed, following Sinai’s approach, as conditional distributions Qzr(⋅|Πn), where Qzr are suitable product measures on the space Π = ⋃nΠn, depending on an auxiliary “free” parameter z = (z1, z2). The transition from (Π, Qzr) to (Πn, Pnr) is based on the asymptotics of the probability Qzr(Πn), furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the Möbius inversion formula and properties of zeroes of the Riemann zeta function.

Article information

Ann. Probab., Volume 39, Number 6 (2011), 2271-2317.

First available in Project Euclid: 17 November 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]
Secondary: 60F05: Central limit and other weak theorems 05D40: Probabilistic methods

Convex lattice polygonal lines limit shape randomization local limit theorem


Bogachev, Leonid V.; Zarbaliev, Sakhavat M. Universality of the limit shape of convex lattice polygonal lines. Ann. Probab. 39 (2011), no. 6, 2271--2317. doi:10.1214/10-AOP607.

Export citation


  • [1] Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.
  • [2] Arratia, R. and Tavaré, S. (1994). Independent process approximations for random combinatorial structures. Adv. Math. 104 90–154.
  • [3] Bárány, I. (1995). The limit shape of convex lattice polygons. Discrete Comput. Geom. 13 279–295.
  • [4] Bellman, R. (1970). Introduction to Matrix Analysis, 2nd ed. McGraw-Hill, New York.
  • [5] Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions. Krieger, Malabar, FL.
  • [6] Bogachev, L. V. and Zarbaliev, S. M. (1999). Limit theorems for a certain class of random convex polygonal lines. Russian Math. Surveys 54 830–832.
  • [7] Bogachev, L. V. and Zarbaliev, S. M. (1999). Approximation of convex functions by random polygonal lines. Dokl. Math. 59 46–49.
  • [8] Bogachev, L. V. and Zarbaliev, S. M. (2004). Approximation of convex curves by random lattice polygons. Preprint, NI04003-IGS, Isaac Newton Inst. Math. Sci., Cambridge. Available at
  • [9] Bogachev, L. V. and Zarbaliev, S. M. (2009). A proof of the Vershik–Prohorov conjecture on the universality of the limit shape for a class of random polygonal lines. Dokl. Math. 79 197–202.
  • [10] Comtet, A., Majumdar, S. N., Ouvry, S. and Sabhapandit, S. (2007). Integer partitions and exclusion statistics: Limit shapes and the largest parts of Young diagrams. J. Stat. Mech. Theory Exp. P10001 (electronic).
  • [11] Dobrushin, R. L. (1970). Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15 458–486.
  • [12] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I, 3rd ed. Wiley, New York.
  • [13] Freiman, G. A. and Granovsky, B. L. (2005). Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws. Trans. Amer. Math. Soc. 357 2483–2507.
  • [14] Freiman, G. A., Vershik, A. M. and Yakubovich, Y. V. (2000). A local limit theorem for random strict partitions. Theory Probab. Appl. 44 453–468.
  • [15] Freiman, G. A. and Yudin, A. A. (2006). The interface between probability theory and additive number theory (local limit theorems and structure theory of set addition). In Representation Theory, Dynamical Systems, and Asymptotic Combinatorics (V. Kaimanovich and A. Lodkin, eds.). Amer. Math. Soc. Transl. Ser. 2 217 51–72. Amer. Math. Soc., Providence, RI.
  • [16] Fristedt, B. (1993). The structure of random partitions of large integers. Trans. Amer. Math. Soc. 337 703–735.
  • [17] Halmos, P. R. (1951). Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea, New York.
  • [18] Hardy, G. H. and Wright, E. M. (1960). An Introduction to the Theory of Numbers, 4th ed. Oxford Univ. Press, Oxford.
  • [19] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge Univ. Press, Cambridge.
  • [20] Ivić, A. (1985). The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications. Wiley, New York.
  • [21] Karatsuba, A. A. (1993). Basic Analytic Number Theory. Springer, Berlin.
  • [22] Khinchin, A. I. (1949). Mathematical Foundations of Statistical Mechanics. Dover, New York.
  • [23] Khinchin, A. Y. (1960). Mathematical Foundations of Quantum Statistics. Graylock Press, Albany, NY.
  • [24] Kolchin, V. F. (1999). Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge.
  • [25] Lancaster, P. (1969). Theory of Matrices. Academic Press, New York.
  • [26] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
  • [27] Prokhorov, Yu. V. (1998). Private communication.
  • [28] Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. Benjamin, New York.
  • [29] Sinai, Ya. G. (1994). Probabilistic approach to the analysis of statistics for convex polygonal lines. Funct. Anal. Appl. 28 108–113.
  • [30] Titchmarsh, E. C. (1952). The Theory of Functions, 2nd ed. Oxford Univ. Press, Oxford.
  • [31] Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function, 2nd ed. Oxford Univ. Press, Oxford.
  • [32] Vershik, A. M. (1994). The limit form of convex integral polygons and related problems. Funct. Anal. Appl. 28 13–20.
  • [33] Vershik, A. M. (1995). Asymptotic combinatorics and algebraic analysis. In Proceedings of the International Congress of Mathematicians (Zürich, 1994) 2 1384–1394. Birkhäuser, Basel.
  • [34] Vershik, A. M. (1996). Statistical mechanics of combinatorial partitions, and their limit configurations. Funct. Anal. Appl. 30 90–105.
  • [35] Vershik, A. M. (1997). Limit distribution of the energy of a quantum ideal gas from the point of view of the theory of partitions of natural numbers. Russian Math. Surveys 52 379–386.
  • [36] Vershik, A. M. (2006). The Kantorovich metric: The initial history and little-known applications. J. Math. Sci. (N. Y.) 133 1410–1417.
  • [37] Vershik, A. and Zeitouni, O. (1999). Large deviations in the geometry of convex lattice polygons. Israel J. Math. 109 13–27.
  • [38] Widder, D. V. (1941). The Laplace Transform. Princeton Mathematical Series 6. Princeton Univ. Press, Princeton, NJ.
  • [39] Yeh, J. (1995). Martingales and Stochastic Analysis. Series on Multivariate Analysis 1. World Scientific, Singapore.
  • [40] Zarbaliev, S. M. (2004). Limit theorems for random convex polygonal lines. Ph.D. thesis. Moscow State Univ., Moscow.