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

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

Published electronically:
July 28, 2008

MathSciNet review:
2434284

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Abstract: We consider a planar Brownian loop that is run for a time and conditioned on the event that its range encloses the unusually high area of , with being large. The conditioned process, denoted by , was proposed by Senya Shlosman as a model for the fluctuation of a phase boundary. We study the deviation of the range of from a circle of radius . This deviation is measured by the inradius and outradius , which are the maximal radius of a disk enclosed by the range of , 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 by at most .

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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