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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Coalescence on the real line


Authors: Paul Balister, Béla Bollobás, Jonathan Lee and Bhargav Narayanan
Journal: Trans. Amer. Math. Soc. 371 (2019), 1583-1619
MSC (2010): Primary 60K35; Secondary 60D05, 60G55
DOI: https://doi.org/10.1090/tran/7391
Published electronically: July 31, 2018
MathSciNet review: 3894028
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Abstract: We study a geometrically constrained coalescence model derived from spin systems. Given two probability distributions $ \mathbb{P}_R$ and $ \mathbb{P}_B$ on the positive reals with finite means, colour the real line alternately with red and blue intervals so that the lengths of the red intervals have distribution $ \mathbb{P}_R$, the lengths of the blue intervals have distribution $ \mathbb{P}_B$, and distinct intervals have independent lengths. Now, iteratively update this colouring of the line by coalescing intervals: change the colour of any interval that is surrounded by longer intervals so that these three consecutive intervals subsequently form a single monochromatic interval. We say that a colour (either red or blue) wins if every point of the line is eventually of that colour. Holroyd, in 2010, asked the following question: under what natural conditions on the initial distributions is one of the colours almost surely guaranteed to win? It turns out that the answer to this question can be quite counter-intuitive due to the non-monotone dynamics of the model. In this paper, we investigate various notions of ``advantage'' one of the colours might initially possess, and in the course of doing so, we determine which of the two colours emerges victorious for various non-trivial pairs of initial distributions.


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Additional Information

Paul Balister
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email: pbalistr@memphis.edu

Béla Bollobás
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom – Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152 – and – London Institute for Mathematical Sciences, 35a South Street, Mayfair, London W1K 2XF, United Kingdom
Email: b.bollobas@dpmms.cam.ac.uk

Jonathan Lee
Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
Email: jonathan.lee@merton.ox.ac.uk

Bhargav Narayanan
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

DOI: https://doi.org/10.1090/tran/7391
Received by editor(s): October 24, 2016
Received by editor(s) in revised form: June 18, 2017
Published electronically: July 31, 2018
Additional Notes: The first and second authors were partially supported by NSF grant DMS-1600742, and the second author also wishes to acknowledge support from EU MULTIPLEX grant 317532.
Article copyright: © Copyright 2018 American Mathematical Society