Journal of Statistical Physics, Vol. 80, Nos. 3/4, 1995, p. 545-563.
Finite one-dimensional random processes with local interaction are presented which keep some information of a topological nature about their initial conditions during time, the logarithm of whose expectation grows asymptotically at least as M^3, where M is the "size" of the set R_M of states of one component. Actually R_M is a circle of length M. At every moment of the discrete time every component turns into some kind of average of its neighbors, after which it makes a random step along this circle. All these steps are mutually independent and identically distributed. In the present version the absolute values of the steps never exceed a constant. The processes are uniform in space, time, and the set of states. This estimation contributes to our awareness of what kind of stable behavior one can expect from one-dimensional random processes with local interaction.