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A Markov Process on Cyclic Words
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The TASEP (totally asymmetric simple exclusion process) studied here is a Markov chain on cyclic words over the alphabet{1,2,...,n} given by at each time step sorting an adjacent pair of letters chosen uniformly at random. For example, from the word 3124 one may go to 1324, 3124, 3124, 4123 by sorting the pair 31, 12, 24, or 43.

Two words have the sametype if they are permutations of each other. If we restrict TASEP to words of some particular type m we get an ergodic Markov chain whose stationary distribution we denote by ζm. Soζm (u) is the asymptotic proportion of time spent in the state u if the chain started in some word of type m. The distribution ζ is the main object of study in this thesis. This distribution turns out to have several remarkable properties, and alternative characterizations. It has previously been studied both from physical, combinatorial, and probabilitistic viewpoints.

In the first chapter we give an extended summary of known results and results in this thesis concerning ζ. The new results are described (and proved) in detail in Papers I - IV.

The new results in Papers I and II include an explicit formula for the value ofζat sorted words and a product formula for decomposable words. We also compute some correlation functions for ζ. In Paper III we study of a generalization of TASEP to Weyl groups. In Paper IV we study a certain scaling limit of ζ, finding several interesting patterns of which we prove some. We also study an inhomogenous version of TASEP, in which different particles get sorted at different rates, which generalizes the homogenous version in several aspects. In the first chapter we compute some correlation functions for ζ

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , vii, 35 p.
Series
TRITA-MAT-A, 2014:12
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-156862ISBN: 978-91-7595-357-1 (print)OAI: oai:DiVA.org:kth-156862DiVA: diva2:768235
Public defence
2014-12-12, E3, Osquars backe 14, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20141204

Available from: 2014-12-04 Created: 2014-12-03 Last updated: 2015-08-27Bibliographically approved
List of papers
1. Stationary probability of the identity for the TASEP on a Ring
Open this publication in new window or tab >>Stationary probability of the identity for the TASEP on a Ring
2012 (English)Other (Other academic)
Abstract [en]

Consider the following Markov chain on permutations of length n. At each time step we choose a random position. If the letter at that position is smaller than the letter immediately to the left (cyclically) then these letters swap positions. Otherwise nothing happens, corresponding to a loop in the Markov chain. This is the circular TASEP. We compute the average proportion of time the chain spends at the identity permutation (and, in greater generality, at sorted words). This answers a conjecture by Thomas Lam.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-156857 (URN)
Note

QC 20141203

Available from: 2014-12-03 Created: 2014-12-03 Last updated: 2014-12-04Bibliographically approved
2. A product formula for the TASEP on a ring
Open this publication in new window or tab >>A product formula for the TASEP on a ring
2016 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 48, no 2, 247-259 p.Article in journal (Refereed) Published
Abstract [en]

For a random permutation sampled from the stationary distributionof the TASEP on a ring, we show that, conditioned on the event that the rstentries are strictly larger than the last entries, the order of the rst entries isindependent of the order of the last entries. The proof uses multi-line queues asdened by Ferrari and Martin, and the theorem has an enumerative combinatorialinterpretation in that setting.As an application we prove a conjecture of Lam and Williams concerningSchubert factors of the stationary probability of certain states.Finally, we present a conjecture for the case where the small and large entriesare not separated.

Place, publisher, year, edition, pages
Wiley-Blackwell, 2016
Keyword
TASEP, exclusion process, multi-line queue, Markov chain
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-156858 (URN)10.1002/rsa.20595 (DOI)000367624300002 ()2-s2.0-84952987496 (Scopus ID)
Note

QC 20160205

Available from: 2014-12-03 Created: 2014-12-03 Last updated: 2017-12-05Bibliographically approved
3. TASEP in any Weyl group
Open this publication in new window or tab >>TASEP in any Weyl group
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We investigate a Markov chain dened by Thomas Lam [6], whichgeneralizes the multi-type TASEP on a ring to any Weyl group. For groups of typeC we dene an analogue of the multiline queues of Ferrari and Martin (which com-pute the stationary distribution for the classical TASEP). While our constructiondoes not suce for nding the stationary distribution, the construction gives thestationary distribution of a certain projection of Lam's chain. Also, our approach isincremental, in the sense that the construction appears to t into a pattern of 'con-jugation matrices', which remains to be fully worked out. We conjecture an explicitformula for the partition function of the model. Finally, we prove a theorem for theclassical TASEP which ts into the picture of viewing TASEP in a permutation-freeway.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-156859 (URN)
Note

QS 2014

Available from: 2014-12-03 Created: 2014-12-03 Last updated: 2014-12-04Bibliographically approved
4. Continuous multiline queues and TASEP
Open this publication in new window or tab >>Continuous multiline queues and TASEP
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-156860 (URN)
Note

QC 20141203

Available from: 2014-12-03 Created: 2014-12-03 Last updated: 2014-12-04Bibliographically approved

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