Please note that updated Statistical Laboratory seminar lists are shown
on our web page at
http://www.statslab.cam.ac.uk/Seminars/statsemeast2000.html
SEMINARS
STATISTICAL LABORATORY
UNIVERSITY OF CAMBRIDGE
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* ALL INTERESTED ARE WELCOME *
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Seminars are in Meeting Room 12, Centre for Mathematical Sciences
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Friday 19 May 2000
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2pm Bob Griffiths (University of Oxford)
THE AGE OF A MUTATION IN A GENERAL BINARY COALESCENT TREE}
Kimura and Ohta showed that the expected age of a neutral
mutation observed to be of frequency $x$ in a population is
$-2x(1-x)^{-1}\log x.$
This classical result is put in a general coalescent process context that
allows questions to be asked about mutations in the sample, as well as in
the population. In the general context coalescence joins have
an exchangeable distribution and coalescence times are have an arbitary
distribution.
Assuming a point mutation it is possible to find the
distribution of the age of a mutation given its frequency in a sample,
or population. It is also of interest to study the characteristics of the
subtree under a mutation.
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Friday 26 May 2000
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2pm David B. Wilson (Microsoft Research)
MIXING TIMES OF LOZENGE TILING AND CARD SHUFFLING MARKOV CHAINS
We introduce a simple technique that can be applied to a number of classes
of Markov chains to obtain upper bounds and lower bounds on their mixing
times, the number of steps the Markov chains take to approach their
equilibrium distribution. One application is to a class of Markov chains
introduced by Luby, Randall, and Sinclair to generate random tilings of
regions by lozenges. The mixing time bounds derived here improve substantially
upon the previous results. In another application we resolve a few
questions raised by Diaconis and Saloff-Coste, by lower bounding the
mixing time of various card shuffling Markov chains. Our lower bounds are
within a constant factor of their upper bounds. When we apply this technique
to modify an analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper
bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear
extensions. Using this same technique we obtain a lower bound on the mixing
time of the random walk on high-dimensional product graphs, which is only
a factor of 1-o(1) smaller than a known upper bound.
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Seminar organizer: Susan Pitts
Please see also the /~grg/seminars/probsem.html - Informal Probability
Seminars.
UNIVERSITY OF CAMBRIDGE
STATISTICAL LABORATORY
CENTRE FOR MATHEMATICAL SCIENCES
WILBERFORCE ROAD
CAMBRIDGE CB3 0WB
Tel: (01223) 337958
Fax: (01223) 337956
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