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Publication 08-CNA-13

Dynamics and Self-Similarity in Min-Driven Clustering

Govind Menon
Division of Applied Mathematics
Box F
Brown University
Providence, RI 02912
Email: menon@dam.brown.edu

Barbara Niethammer
Mathematical Institute
University of Oxford
Oxford, OX1 3LB, UK
Email: niethammer@maths.ox.ac.uk

Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
Email: rpego@cmu.edu

 

Abstract: We study a mean-field model for a clustering process that may be described informally as follows. At each step a random integer $k$ is chosen with probability $p_k$, and the smallest cluster merges with $k$ randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in $(0,\infty)$, and we establish necessary and sufficient conditions for approach to self-similar form. We also characterize eternal solutions for this model via a Lévy-Khintchine formula. The analysis is based on an explicit solution formula discovered by Gallay and Mielke, extended using a careful choice of time scale.

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