for all $j$ large enough. On the other hand, it is easy to check that all remaining configurations for $v(t_j)$ are possible and stable, that is, once you reach them, you stay in them forever.

Call a configuration $\bar v$ *admissible*, if there is a collection $\xi_1$, $\xi_2$, …, $\xi_M$ such that the system (\[eq:uxisym\]) has a solution for some $u=(u_1,\dots,u_M)$ having type $\bar v$. If $M$ is not divisible by $3$, according to Proposition \[prop\_sym1\] all configurations $\bar v$ are admissible. On the other hand, it is easy to see that if $M=3K$ then a necessary and sufficient condition for a non-zero configuration $\bar v$ to be admissible is |v\_i=&& i 3=0,\
|v\_j=&& j 3=1,\
|v\_k=&& k 3=2. This establishes all possible stable configurations for $v$ and hence the potential $u$, thus determining the subset of $\M$ where points are adsorbed for sufficiently large times, namely, $\xi_i(t)\to\infty$ if and only if $v_i(t_j)=0$ for all large $j$.

Moreover, whenever we see a subsequence of type $(v_{k-1},v_k,v_{k+1})=(\ast,0,\ast)$, we have $$0\le \lim_{j\to\infty} \left[t_j-u_k(t_j)\right]<\infty,$$ and for a subsequence of type $(v_{k-1},v_k,v_{k+1},v_{k+2})=(\ast,0,0,\ast)$ we have $$\lim_{j\to\infty} \frac{u_k(t_j)}{t_j}
 =\lim_{j\to\infty} \frac{u_{k+1}(t_j)}{t_j}=\frac 12$$ by the strong law. Setting $$\a=\frac 1{\lim_{j\to\infty} |\{i\in\M:\ v_i(t_j)>0,
v_{i+1}(t_j)=0\}|}$$ finishes the proof of the 