2012/03/01 by Ferrari, Pablo A., Martin, James B.
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1203.0176
We consider "bridges" for the simple exclusion process on Z, either symmetric or asymmetric, in which particles jump to the right at rate p and to the left at rate 1-p. The initial state O has all negative sites occupied and all non-negative sites empty. We study the probability that the process is again in state O at time t, and the behaviour of the process on [0,t] conditioned on being in state O at time t. In the case p=1/2, we find that such a bridge typically goes a distance of order t (in the sense of graph distance) from the initial state. For the asymmetric systems, we note an interesting duality which shows that bridges with parameters p and 1-p have the same distribution; the maximal distance of the process from the original state behaves like c(p)log(t) for some constant c(p) depending on p. (For p>1/2, the front particle therefore travels much less far than the bridge of the corresponding random walk, even though in the unconditioned process the path of the front particle dominates a random walk.) We mention various further questions.