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

Full-text PDF Free Access

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 $\textrm {R}_\textrm {in}(X)$ and outradius $\textrm {R}_\textrm {out}(X)$, 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 $T^{2/3 + \epsilon }$.

- Kenneth S. Alexander,
*Cube-root boundary fluctuations for droplets in random cluster models*, Comm. Math. Phys.**224**(2001), no. 3, 733–781. MR**1871907**, DOI https://doi.org/10.1007/s220-001-8022-2 - 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** - 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** - Richard Durrett,
*Probability: theory and examples*, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR**1609153** - Ioannis Karatzas and Steven E. Shreve,
*Brownian motion and stochastic calculus*, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1988. MR**917065** - 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** - 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**, DOI https://doi.org/10.1007/s00440-003-0276-0 - D. W. Stroock,
*An introduction to the theory of large deviations*, Universitext, Springer-Verlag, New York, 1984. MR**755154** - David Williams,
*Probability with martingales*, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR**1155402** - G. Wulff. Zur Frage der Geschwingkeit des Wachstums und der Auflosung der Krystallflachen.
*Z. Kryst.*, 34:449–530, 1901.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
60J65,
60F10

Retrieve articles in all journals with MSC (2000): 60J65, 60F10

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

MR Author ID:
137920

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