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January 1999 Existence of Hydrodynamics for the Totally Asymmetric Simple K-Exclusion Process
Timo Seppäläinen
Ann. Probab. 27(1): 361-415 (January 1999). DOI: 10.1214/aop/1022677266


In a totally asymmetric simple $K$-exclusion process, particles take nearest-neighbor steps to the right on the lattice Z, under the constraint that each site contain at most $K$ particles. We prove that such processes satisfy hydrodynamic limits under Euler scaling, and that the limit of the empirical particle profile is the entropy solution of a scalar conservation law with a concave flux function. Our technique requires no knowledge of the invariant measures of the process, which is essential because the equilibria of asymmetric $K$-exclusion are unknown. But we cannot calculate the flux function precisely. The proof proceeds via a coupling with a growth model on the two-dimensional lattice. In addition to the basic $K$-exclusion with constant exponential jump rates, we treat the sitedisordered case where each site has its own jump rate, randomly chosen but frozen for all time. The hydrodynamic limit under site disorder is new even for the simple exclusion process (the case $K = 1$). Our proof makes no use of the Markov property, so at the end of the paper we indicate how to treat the case with arbitrary waiting times.


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Timo Seppäläinen. "Existence of Hydrodynamics for the Totally Asymmetric Simple K-Exclusion Process." Ann. Probab. 27 (1) 361 - 415, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0947.60088
MathSciNet: MR1681094
Digital Object Identifier: 10.1214/aop/1022677266

Primary: 60K35
Secondary: 60K25 , 82C22 , 82C24

Keywords: $ M/M/1/K$ queues in series , $K$-exclusion process , Growth model , Hydrodynamic limit , marching soldiers model , Quenched disorder , subadditive process

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 1 • January 1999
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