Header

UZH-Logo

Maintenance Infos

A two-time-scale phenomenon for a fragmentation-coagulation process


Bertoin, J (2010). A two-time-scale phenomenon for a fragmentation-coagulation process. Electronic Communications in Probability, 15:253-262.

Abstract

Consider two urns, A and B, where initially A contains a large number n of balls and B is empty. At each step, with equal probability, either we pick a ball at random in A and place it in B, or vice-versa (provided of course that A, or B, is not empty). The number of balls in B after n steps is of order n√, and this number remains essentially the same after n√ further steps. Observe that each ball in the urn B after n steps has a probability bounded away from 0 and 1 to be placed back in the urn A after n√ additional steps. So, even though the number of balls in B does not evolve significantly between n and n+n√, the precise contain of urn B does.
This elementary observation is the source of an interesting two-time-scale phenomenon which we illustrate using a simple model of fragmentation-coagulation. Inspired by Pitman's construction of coalescing random forests, we consider for every n∈N a uniform random tree with n vertices, and at each step, depending on the outcome of an independent fair coin tossing, either we remove one edge chosen uniformly at random amongst the remaining edges, or we replace one edge chosen uniformly at random amongst the edges which have been removed previously. The process that records the sizes of the tree-components evolves by fragmentation and coagulation. It exhibits subaging in the sense that when it is observed after k steps in the regime k∼tn+sn√ with t>0 fixed, it seems to reach a statistical equilibrium as n→∞; but different values of t yield distinct pseudo-stationary distributions.

Abstract

Consider two urns, A and B, where initially A contains a large number n of balls and B is empty. At each step, with equal probability, either we pick a ball at random in A and place it in B, or vice-versa (provided of course that A, or B, is not empty). The number of balls in B after n steps is of order n√, and this number remains essentially the same after n√ further steps. Observe that each ball in the urn B after n steps has a probability bounded away from 0 and 1 to be placed back in the urn A after n√ additional steps. So, even though the number of balls in B does not evolve significantly between n and n+n√, the precise contain of urn B does.
This elementary observation is the source of an interesting two-time-scale phenomenon which we illustrate using a simple model of fragmentation-coagulation. Inspired by Pitman's construction of coalescing random forests, we consider for every n∈N a uniform random tree with n vertices, and at each step, depending on the outcome of an independent fair coin tossing, either we remove one edge chosen uniformly at random amongst the remaining edges, or we replace one edge chosen uniformly at random amongst the edges which have been removed previously. The process that records the sizes of the tree-components evolves by fragmentation and coagulation. It exhibits subaging in the sense that when it is observed after k steps in the regime k∼tn+sn√ with t>0 fixed, it seems to reach a statistical equilibrium as n→∞; but different values of t yield distinct pseudo-stationary distributions.

Statistics

Citations

Altmetrics

Downloads

17 downloads since deposited on 24 Apr 2013
5 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2010
Deposited On:24 Apr 2013 11:33
Last Modified:05 Apr 2016 16:43
Publisher:Institute of Mathematical Statistics
ISSN:1083-589X
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/ECP.v15-1552
Related URLs:http://arxiv.org/abs/1001.3721

Download

Preview Icon on Download
Preview
Filetype: PDF
Size: 135kB
View at publisher

TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.

Author Collaborations