The jammed phase of the biham-middleton-levine traffic model

Initially a car is placed with probability p at each site of the two-dimensional integer lattice. Each car is equally likely to be East-facing or North-facing, and different sites receive inde-pendent assignments. At odd time steps, each North-facing car moves one unit North if there is a vacant site for it to move into. At even time steps, East-facing cars move East in the same way. We prove that when p is sufficiently close to 1 tra�c is jammed, in the sense that no car moves infinitely many times. The result extends to several variant settings, including a model with cars moving at random times, and higher dimensions. © 2005 Applied Probability Trust.