The expected area of the filled planar Brownian loop is \(\pi/5\). (English) Zbl 1107.82023
Summary: Let \(B_{t}\), \(0\leq t \leq 1\) 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 the complement of \(B [0,1]\). We show that the expected area of this hull is \(\pi /5\). The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). As a consequence of this result, using Yor’s formula 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 \(\pi/30\); this value could not be obtained directly using Yor’s index description.
MSC:
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 60J65 | Brownian motion |
References:
| [1] | Cardy, Phys. Rev. Lett., 72, 1580 (1994) · Zbl 0973.82507 · doi:10.1103/PhysRevLett.72.1580 |
| [2] | Comtet, J. Phys. A: Math. Gen., 23, 3563 (1990) · Zbl 0714.60066 · doi:10.1088/0305-4470/23/15/027 |
| [3] | Lawler, G.F.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114, Providence, RI: Amer. Math. Soc., 2005 · Zbl 1074.60002 |
| [4] | Lawler, Probab. Theory Related Fields, 128, 565 (2004) · Zbl 1049.60072 · doi:10.1007/s00440-003-0319-6 |
| [5] | Lawler, The chordal case. J. Amer. Math. Soc., 16, 917 (2003) · Zbl 1030.60096 · doi:10.1090/S0894-0347-03-00430-2 |
| [6] | Lawler, Math. Res. Lett., 8, 401 (2001) · Zbl 1114.60316 |
| [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 · Zbl 1069.60089 |
| [8] | Richard, J. Phys. A., 37, 4493 (2004) · Zbl 1047.82012 · doi:10.1088/0305-4470/37/16/002 |
| [9] | Thacker, J.: Hausdorff Dimension of the Brownian Loop Soup. In preparation (2005) |
| [10] | Schramm, Electron. J. Probab. Vol., 7, 1 (2) |
| [11] | Vervaat, Ann. Probab., 7, 143 (1979) · Zbl 0392.60058 |
| [12] | Werner, Probability Theory and Related Fields, 99, 111 (1994) · Zbl 0799.60074 · doi:10.1007/BF01199592 |
| [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 · Zbl 1057.60078 |
| [14] | Werner, W.: Conformal restriction and related questions. http://arxiv.org/list/math.PR/0307353, 2003 · Zbl 1189.60032 |
| [15] | Werner, C. R. Acad. Sci. Paris Ser. I Math., 337, 481 (2003) · Zbl 1029.60085 |
| [16] | Werner, W.: The conformally invariant measure on self-avoiding loops. http://arxiv.org/list/math.PR/ 0511605, 2005 |
| [17] | Yor, Z. Wahrsch. Verw. Gebiete, 53, 71 (1980) · Zbl 0436.60057 · doi:10.1007/BF00531612 |
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