Simple exclusion process
introduction
- Heisenberg spin chain model
- can be viewed as a exclusion process (time evolution)
Bethe Ansatz and Exclusion Processes http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=39328&date=2011-02-01
Stochastic growth models in the plane
For aimple case, consider a graph of a random height function h.
Consider the rescaling
h^{\epsion}(x,t)=\epsilon h(\frac{x}{\epsilon},\frac{t}{\epsilon})
Then we expect to have
After some scaling argument, one may use KPZ equation to justify \epsilon^{2/3} as the order og the fluctuations of the above problem. But what is the law of the random \eta ?
Perhaps we can locate an example for which we can find exact formula for h as a result a formula for \eta. So for we have two examples that are "exactly solvable"
These examples are
Hammersley-Aldous-Diaconis (HAD) process and simple exclusion processes.
For the latter a trick known on Bethe ansatz is used to find very explicit formulas for various quantities of interest.
exclusion process
particles jumping from left ro right or from right ro left with given probabilityes p and q (p+q=1)
G(x,t) = probability (x(t)=x | x(0) is distrbuted according to g(x) )
Bethe ansatz
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
articles
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
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- http://ncatlab.org/nlab/show/HomePage
experts on the field