zeros at odd sites. Besides, observe that the sum of the potentials of the even sites equals the sum of the potentials of the odd sites (see Proposition \[prop:even\]); therefore the difference between the total queue to the even sites, and the total [queue]{} to the odd ones, behaves similarly to the zero-mean random walk. It means that there are rather long periods of time during which much longer queues are observed at the even sites in comparison with the queues at the odd sites, and vice versa. Thus, in the case of the asymmetric interaction we observe in the limit $t{\rightarrow}\infty$ a ‘comb pattern’ when $M$ is even, and a ‘flat pattern’ when $M$ is odd.

The picture is completely different in the symmetric case. The Markov chain $v(t)$ is transient for any $M$; moreover, there can be only finitely many paths along which the chain escapes to infinity. By this we mean that if the chain reaches a state belonging to a particular *escape path*, then it will never leave it and will go to infinity along this path, and we will show that there can be only a finite number of limit patterns. An escape path can 