| author: | Nathanaël Berestycki and Rick Durrett |
|---|---|
| title: | A phase transition in the random transposition random walk |
| keywords: | random transposition, random graphs, phase transition, coagulation-fragmentation |
| abstract: |
Our work is motivated by Bourque-Pevzner's simulation study
of the effectiveness of the parsimony method in studying
genome rearrangement, and leads to a surprising result
about the random transposition walk in continuous time on
the group of permutations on n elements starting from the
identity. Let
D
be the minimum number of transpositions needed to go
back to the identity element from the location at time
t
t
.
D
undergoes a phase transition: for
t
0 < c ≤ 1
, the distance
D
, i.e., the distance increases linearly with time;
for
cn/2
~ cn/2
c > 1
,
D
where
cn/2
~ u(c)n
u
is an explicit function satisfying
u(x)<x/2
. Moreover we describe the fluctuations of
D
about its mean at each of the three stages
(subcritical, critical and supercritical). The techniques
used involve viewing the cycles in the random permutation
as a coagulation-fragmentation process and relating the
behavior to the Erdős-Rényi random graph model.
cn/2
|
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| reference: | Nathanaël Berestycki and Rick Durrett (2003), A phase transition in the random transposition random walk, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 17-26 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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| pdf-source: | dmAC0102.pdf (116 K) |
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