Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 813–858 | Cite as

Strong law of large numbers for the capacity of the Wiener sausage in dimension four

  • Amine Asselah
  • Bruno SchapiraEmail author
  • Perla Sousi


We prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four, where a logarithmic correction appears in the scaling. The main step of the proof is to obtain precise asymptotics for the expected value of the capacity. This requires a delicate analysis of intersection probabilities between two independent Wiener sausages.


Capacity Wiener sausage Law of large numbers 

Mathematics Subject Classification

Primary 60F05 60G50 



We warmly thank the referee for his/her very careful reading and insightful comments, which greatly helped improve, clarify and correct the paper.


  1. 1.
    Aizenman, M.: The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory. Commun. Math. Phys. 97(1–2), 91–110 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albeverio, S., Zhou, X.Y.: Intersections of random walks and Wiener sausages in four dimensions. Acta Appl. Math. 45(2), 195–237 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asselah, A., Schapira, B., Sousi, P.: Capacity of the range of random walk on \({\mathbb{Z}}^d\). (2016). arXiv:1602.03499v1
  4. 4.
    Asselah, A., Schapira, B., Sousi, P.: Capacity of the range of random walk on \({\mathbb{Z}}^4\). (2016). arXiv:1611.04567
  5. 5.
    Chung, F., Lu, L.: Concentration inequalities and martingale inequalities: a survey. Internet Math. 3(1), 79–127 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dvoretzky, A., Erdös, P., Kakutani, S.: Double points of paths of Brownian motion in \(n\)-space. Acta Sci. Math. Szeged 12(Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B), 75–81 (1950)Google Scholar
  7. 7.
    Erdös, P., Taylor, S.J.: Some intersection properties of random walk paths. Acta Math. Acad. Sci. Hung. 11, 231–248 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erhard, D., Poisat, J.: Asymptotics of the critical time in Wiener sausage percolation with a small radius. ALEA Lat. Am. J Probab. Math. Stat. 13(1), 417–445 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Getoor, R.K.: Some asymptotic formulas involving capacity. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4(248–252), 1965 (1965)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Khoshnevisan, D.: Intersections of Brownian motions. Expos. Math. 21(2), 97–114 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lawler, G.F.: The probability of intersection of independent random walks in four dimensions. Commun. Math. Phys. 86(4), 539–554 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lawler, G.F.: Intersections of Random Walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA (1991)CrossRefzbMATHGoogle Scholar
  13. 13.
    Le Gall, J.-F.: Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection. Commun. Math. Phys. 104(3), 471–507 (1986)CrossRefzbMATHGoogle Scholar
  14. 14.
    Le Gall, J.-F.: Sur la saucisse de Wiener et les points multiples du mouvement brownien. Ann. Probab. 14(4), 1219–1244 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Le Gall, J.-F.: Fluctuation results for the Wiener sausage. Ann. Probab. 16(3), 991–1018 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Le Gall, J.-F.: Some properties of planar Brownian motion. In: École d’Été de Probabilités de Saint-Flour XX—1990, volume 1527 of Lecture Notes in Mathematics, pp. 111–235. Springer, Berlin (1992)Google Scholar
  17. 17.
    Lieb, E.H., Loss, M.: Analysis, volume 14 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  18. 18.
    Mörters, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pemantle, R., Peres, Y., Shapiro, J.W.: The trace of spatial Brownian motion is capacity-equivalent to the unit square. Probab. Theory Relat. Fields 106(3), 379–399 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Peres, Y.: Intersection-equivalence of Brownian paths and certain branching processes. Commun. Math. Phys. 177(2), 417–434 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (1999)Google Scholar
  22. 22.
    van den Berg, M., Bolthausen, E., den Hollander, F.: Torsional rigidity for regions with a Brownian boundary. (2016). arXiv:1604.07007

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Paris-Est CréteilCréteilFrance
  2. 2.CNRS, Centrale Marseille, I2M, UMR 7373Aix-Marseille UniversitéMarseilleFrance
  3. 3.University of CambridgeCambridgeUK

Personalised recommendations