  We investigate the asymptotic behaviour of a class of self-interacting
nearest neighbour random walks on the one-dimensional integer lattice which are
pushed by a particular linear combination of their own local time on edges in
the neighbourhood of their current position. We prove that in a range of the
relevant parameter of the model such random walkers can be eventually confined
to a finite interval of length depending on the parameter value. The phenomenon
arises as a result of competing self-attracting and self-repelling effects
where in the named parameter range the former wins.
