DIMACS Focus on Discrete Probability Seminar


Non-ergodicity and related phenomena in interacting random processes


Andre Toom


CoRE Building, Room 431
Busch Campus, Rutgers University.


3:30 - 4:30 PM
Wednesday, May 21, 1997

At every point of Z^d there is a component which updates its state at every moment of discrete time depending on the states of its neighbors at the previous time and random noise. Even if every component has only two possible states, 0 and 1, we have many phenomena which are not easy to prove. One of these is the non-uniqueness of invariant measure (called non-ergodicity here) in the NEC-voting and similar models. In the NEC-voting d=2 and a point (i,j)`s neighbors are (i,j+1), (i+1,j), (i,j). (North, East, Center). The point`s state at the next moment is that in which the majority of its neighbors were at the previous moments with probability 1-epsilon and the other state with probability epsilon. If epsilon is small enough, the system has at least two different invariant measures. Similar methods work for systems in which the set of states of each point is the set of integer numbers. For example, let a point`s state at time t+1 be the arithmetical mean of the same neighbors at time t, rounded to the nearest integer, plus random noise. This system displays "pinning": if the probability of noise is small enough, the system's average remains bounded. (Rounding is essential!)

Document last modified on May 20, 1997