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Transactions of the American Mathematical Society

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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}$.


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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

DOI: https://doi.org/10.1090/S0002-9947-08-04366-3
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