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Advances in AnalysisThe Legacy of Elias M. Stein$
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Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger

Print publication date: 2014

Print ISBN-13: 9780691159416

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159416.001.0001

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Internal DLA for Cylinders

Internal DLA for Cylinders

(p.189) Chapter Eight Internal DLA for Cylinders
Advances in Analysis

David Jerison

Lionel Levine

Scott Sheffield

, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger
Princeton University Press

This chapter discusses the continuum limit of internal Diffusion-Limited Aggregation (DLA), a random lattice growth model governed by a deterministic fluid flow equation known as Hele-Shaw flow. The internal DLA model was introduced in 1986 by Meakin and Deutch to describe chemical processes such as electropolishing, etching, and corrosion. The chapter focuses primarily on fluctuations, and seeks to prove the analogous results for the lattice cylinder. In the case of the cylinder, the fluctuations are described in terms of the Gaussian Free Field exactly. The main tools used in the proofs are martingales. As the chapter shows, the martingale property in this context is the counterpart in probability theory of well-known conservation laws for Hele-Shaw flow.

Keywords:   internal Diffusion-Limited Aggregation, internal DLA, lattice cylinder, fluctuations, Gaussian Free Field, martingales, Hele-Shaw flow

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