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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Fluctuation of a planar Brownian loop capturing a large area

Authors: Alan Hammond and Yuval Peres
Journal: Trans. Amer. Math. Soc. 360 (2008), 6197-6230
MSC (2000): Primary 60J65; Secondary 60F10
Published electronically: July 28, 2008
MathSciNet review: 2434284
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a planar Brownian loop $ B$ that is run for a time $ T$ and conditioned on the event that its range encloses the unusually high area of $ \pi T^2$, with $ T \in (0,\infty)$ being large. The conditioned process, denoted by $ X$, was proposed by Senya Shlosman as a model for the fluctuation of a phase boundary. We study the deviation of the range of $ X$ from a circle of radius $ T$. This deviation is measured by the inradius $ {\rm R}_{\rm in}(X)$ and outradius $ {\rm R}_{\rm out}(X)$, which are the maximal radius of a disk enclosed by the range of $ X$, and the minimal radius of a disk that contains this range. We prove that, in a typical realization of the conditioned measure, each of these quantities differs from $ T$ by at most $ T^{2/3 + \epsilon}$.

References [Enhancements On Off] (What's this?)

  • 1. Kenneth S. Alexander, Cube-root boundary fluctuations for droplets in random cluster models, Comm. Math. Phys. 224 (2001), no. 3, 733–781. MR 1871907, 10.1007/s220-001-8022-2
  • 2. Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419
  • 3. Amir Dembo and Ofer Zeitouni, Large deviations and applications, Handbook of stochastic analysis and applications, Statist. Textbooks Monogr., vol. 163, Dekker, New York, 2002, pp. 361–416. MR 1882715
  • 4. Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153
  • 5. Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1988. MR 917065
  • 6. L. A. Santaló, Integral geometry, Studies in Global Geometry and Analysis, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1967, pp. 147–193. MR 0215272
  • 7. Hasan B. Uzun and Kenneth S. Alexander, Lower bounds for boundary roughness for droplets in Bernoulli percolation, Probab. Theory Related Fields 127 (2003), no. 1, 62–88. MR 2006231, 10.1007/s00440-003-0276-0
  • 8. S. R. S. Varadhan, Large deviations and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 46, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. MR 758258
  • 9. David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR 1155402
  • 10. G. Wulff.
    Zur Frage der Geschwingkeit des Wachstums und der Auflosung der Krystallflachen.
    Z. Kryst., 34:449-530, 1901.

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Additional Information

Alan Hammond
Affiliation: Department of Mathematical Sciences, New York University-Courant Institute, 251 Mercer Street, New York, New York 10012-1185

Yuval Peres
Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052

Received by editor(s): February 3, 2006
Received by editor(s) in revised form: June 3, 2006
Published electronically: July 28, 2008
Additional Notes: The research of the second author was supported in part by NSF grants #DMS-0244479 and #DMS-0104073
Article copyright: © Copyright 2008 Alan Hammond and Yuval Peres