A discrete time interactive exclusive random walk of infinitely many particles on one-dimensional lattices (Q756861)

Stationary measures for a discrete time interactive exclusive random walk of infinitely many particles on \({\mathbb{Z}}\) are considered. The process is a Markov process on the configuration space \({\mathcal X}=\{0,1\}^{{\mathbb{Z}}}\) such that \(\eta \equiv (...\eta_{-1}\eta_ 0\eta_ 1...)\in {\mathcal X}\) at time t changes randomly to \(\eta'\equiv (...\eta'_{-1}\eta'_ 0\eta'_ 1...)\in {\mathcal X}\) at time \(t+1\) by the following rule: For each edge \((i,i+1)\), \(i\in {\mathbb{Z}}\), an element \(\omega_ i\) of \(\{e,\bar e\}\) is chosen independently with \(Prob\{\omega_ i=e\}=\alpha\) if \(\eta_ i\neq \eta_{i+1}\) and with \(Prob\{\omega_ i=e\}=\beta\) if \(\eta_ i=\eta_{i+1}\). Then \(\eta'_ i\eta'_{i+1}= \eta_{i+1}\eta_ i\) if \(\omega_{i-1}\omega_ i\omega_{i+1}= \bar ee\bar e\) and \(\eta'_ i\eta'_{i+1}=\eta_ i\eta_{i+1}\) otherwise. It is shown that after a simple limiting procedure a well-known simple exclusion process is obtained and that the set of the stationary measures for the process is equal to the set of canonical Gibbs states with the nearest neighbor potential \(-kT \log\gamma\), \(\gamma=[(1-\alpha)/(1- \beta)]^ 2,\) and hence the structrure of stationary measures is completely known.