  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.
