Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Windings of planar random walks and averaged Dehn function

Bruno Schapira and Robert Young

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Abstract

We prove sharp estimates on the expected number of windings of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.

Résumé

Le principal résultat de cet article donne un équivalent précis de l’espérance du nombre total de tours effectués par la marche aléatoire simple sur ℤ2 ou sur le réseau triangulaire. Comme corollaire, nous obtenons une nouvelle borne inférieure de la fonction de Dehn moyennée sur ℤd, d ≥ 2, qui mesure l’aire moyenne du disque remplissant de manière optimale une courbe de longueur donnée.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 130-147.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170233

Digital Object Identifier
doi:10.1214/10-AIHP365

Mathematical Reviews number (MathSciNet)
MR2779400

Zentralblatt MATH identifier
05864078

Subjects
Primary: 52C45: Combinatorial complexity of geometric structures [See also 68U05] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Simple random walk Winding number Averaged Dehn function

Citation

Schapira, Bruno; Young, Robert. Windings of planar random walks and averaged Dehn function. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 130--147. doi:10.1214/10-AIHP365. https://projecteuclid.org/euclid.aihp/1294170233


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References

  • [1] G. Baumslag, C. F. Miller III and H. Short. Isoperimetric inequalities and the homology of groups. Invent. Math. 113 (1993) 531–560.
  • [2] C. Bélisle. Windings of random walks. Ann. Probab. 17 (1989) 1377–1402.
  • [3] O. Bogopolski and E. Ventura. The mean Dehn functions of abelian groups. J. Group Theory 11 (2008) 569–586.
  • [4] I. S. Borisov. On the rate of convergence in the “conditional” invariance principle. Teor. Verojatn. Primen. 23 (1978) 67–79.
  • [5] M. R. Bridson. The geometry of the word problem. In Invitations to Geometry and Topology. Oxf. Grad. Texts Math. 7 29–91. Oxford Univ. Press, Oxford, 2002.
  • [6] U. Einmahl. Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 (1989) 20–68.
  • [7] C. Garban and J. A. Trujillo Ferreras. The expected area of the filled planar Brownian loop is π / 5. Comm. Math. Phys. 264 (2006) 797–810.
  • [8] M. Gromov. Asymptotic invariants of infinite groups. In Geometric Group Theory, Vol. 2 (Sussex, 1991) 1–295. London Math. Soc. Lecture Note Ser. 182. Cambridge Univ. Press, Cambridge, 1993.
  • [9] J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent RV’s and the sample DF, I. Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131.
  • [10] J.-F. Le Gall. Some properties of planar Brownian motion. In École d’Été de Probabilités de Saint-Flour XX, 1990 111–235. Lecture Notes in Math. 1527. Springer, Berlin, 1992.
  • [11] P. Lévy. Processus Stochastiques et Mouvement Brownien. Suivi d’une note de M. Loève. Gauthier-Villars, Paris, 1948.
  • [12] S. C. Port and C. J. Stone. Brownian Motion and Classical Potential Theory. Academic Press, New York, 1978.
  • [13] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1999.
  • [14] F. Spitzer. Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 (1958) 187–197.
  • [15] F. Spitzer. Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrsch. Verw. Gebiete 3 (1964) 110–121.
  • [16] W. Werner. Sur les points autour desquels le mouvement Brownien plan tourne beaucoup. Probab. Theory Related Fields 99 (1994) 111–144.
  • [17] M. Yor. Loi de l’indice du lacet Brownien, et distribution de Hartman–Watson. Z. Wahrsch. Verw. Gebiete 53 (1980) 71–95.
  • [18] R. Young. Averaged Dehn functions for nilpotent groups. Topology 47 (2008) 351–367.
  • [19] A. Y. Zaitsev. Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments. ESAIM Probab. Statist. 2 (1998) 41–108 (electronic).