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The Expected Area of the Filled Planar Brownian Loop is π/5

Abstract

Let B t ,0≤t≤1 be a planar Brownian loop (a Brownian motion conditioned so that B 0=B 1). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of \B[0,1]. We show that the expected area of this hull is π/5. The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). As a consequence of this result, using Yor's formula [17] for the law of the index of a Brownian loop, we find that the expected area of the region inside the loop having index zero is π/30; this value could not be obtained directly using Yor's index description.

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References

  1. Cardy, J.: Mean area of self-avoiding loops. Phys. Rev. Lett. 72, 1580–1583 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Comtet, A., Desbois, J., Ouvry, S.: Winding of planar Brownian curves. J. Phys. A: Math. Gen. 23, 3563–3572 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Lawler, G.F.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114, Providence, RI: Amer. Math. Soc., 2005

  4. Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Related Fields 128, 565–588 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lawler, G.F., Schramm O., Werner, W.: Conformal restriction. The chordal case. J. Amer. Math. Soc. 16, 917–955 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Lawler, G.F., Schramm, O., Werner, W.: The dimension of the planar Brownian frontier is 4/3. Math. Res. Lett. 8, 401–411 (2001)

    MATH  MathSciNet  Google Scholar 

  7. Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Fractal geometry and applications, A jubilee of Benoît Mandelbrot, Proc. Symp. Pure Math. 72, Providence, RI: Amer. Math. Soc., 2004

  8. Richard, C.: Area distribution of the planar random loop boundary. J. Phys. A. 37, 4493–4500 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Thacker, J.: Hausdorff Dimension of the Brownian Loop Soup. In preparation (2005)

  10. Schramm, O.: A percolation formula. Electron. J. Probab. Vol. 7(2), 1–13 (2001)

    Google Scholar 

  11. Vervaat, W.: A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7, 143–149 (1979)

    MATH  MathSciNet  Google Scholar 

  12. Werner, W.: Sur l'ensemble des points autour desquels le mouvement brownien plan tourne beaucoup. Probability Theory and Related Fields 99, 111–142 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Werner, W.: Random planar curves and Schramm-Loewner evolutions. In: Lecture Notes from the 2002 Saint-Flour Summer School, L.N. Math. 1840, Berlin-Heidelberg-New York Springer, 2004, pp. 107–195

  14. Werner, W.: Conformal restriction and related questions. http://arxiv.org/list/math.PR/0307353, 2003

  15. Werner, W.: SLEs as boundaries of clusters of Brownian loops. C. R. Acad. Sci. Paris Ser. I Math. 337, 481–486 (2003)

    MATH  Google Scholar 

  16. Werner, W.: The conformally invariant measure on self-avoiding loops. http://arxiv.org/list/math.PR/ 0511605, 2005

  17. Yor, M.: Loi de l'indice du lacet brownien, et distribution de Hartman-Watson. Z. Wahrsch. Verw. Gebiete 53, 71–95 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to José A. Trujillo Ferreras.

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Communicated by M. Aizenman

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Garban, C., Ferreras, J. The Expected Area of the Filled Planar Brownian Loop is π/5. Commun. Math. Phys. 264, 797–810 (2006). https://doi.org/10.1007/s00220-006-1555-2

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  • DOI: https://doi.org/10.1007/s00220-006-1555-2

Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Hull
  • Brownian Motion