How to squeeze the toothpaste back into the tube

Abstract

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.