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

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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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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    Christophe Garban & José A. Trujillo Ferreras

  2. Ecole Normale Superieure, 45 rue d'ulm, 75230, Paris Cedex 05, France

    Christophe Garban

Authors
  1. Christophe Garban
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  2. José A. Trujillo Ferreras
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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

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