Abstract
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 asM 3, whereM is the “size” of the setR M of states of one component. ActuallyR M is a circle of lengthM. 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.
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Partially supported by NSF grant #DMS-932 1216.
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Toom, A. Simple one-dimensional interaction systems with superexponential relaxation times. J Stat Phys 80, 545–563 (1995). https://doi.org/10.1007/BF02178547
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DOI: https://doi.org/10.1007/BF02178547