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A probabilistic model for interfaces in a martensitic phase transition

Part of: Markov processes Applications to specific types of physical systems

Published online by Cambridge University Press:  09 August 2022

Pierluigi Cesana  and
Ben M. Hambly
Pierluigi Cesana*
Affiliation:
Kyushu University
Ben M. Hambly*
Affiliation:
University of Oxford
*
*Postal address: Institute of Mathematics for Industry, 744 Motooka, Fukuoka 819-0395, Japan. Email address: cesana@math.kyushu-u.ac.jp
**Postal address: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK. Email address: hambly@maths.ox.ac.uk
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Abstract

We analyse features of the patterns formed from a simple model for a martensitic phase transition that fragments the unit square into rectangles. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the lengths of the interfaces in the pattern, and we establish limit theorems for some of the asymptotics of the interface profile. In particular, we are able to use a general branching process to show almost sure power law decay of the number of interfaces of at least a certain size and a general branching random walk to examine the numbers of rectangles of a certain aspect ratio. In doing so we extend a theorem on the growth of the general branching random walk as well as developing results on the tail behaviour of the limiting random variable in our general branching process.

Keywords

Martensitegeneral branching processgeneral branching random walkpower laws

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 82D35: Metals
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1081 - 1105
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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