case describes the [allocation process]{} in which a particle is allocated with equal probabilities to one of the sites with maximal potential. The asymptotic behaviour (as $t{\rightarrow}\infty$) of this limit process is briefly discussed in Section \[binf\].

Results {#res}
-------

\[thm:A2\_b0\] Suppose $U_i=\{i,i+1\}$, $i\in\M$. Then, with probability $1$, there is a $t_0=t_0(\omega)$ (depending also on the initial configuration) such that for all $t\ge t_0$ \[eq\_at\_most2\] |\_i(t)-\_[i+2]{}(t)|2 for $i\in\M$. Moreover, \[eq\_xieo\] \_i(t)=t[M]{} + \_i(t)+{

[rc]{} 0, &,\
\
(-1)\^i Z(t), &,

. where $|\eta_i(t)|\le 2M$ and for some $\sigma>0$ $$\lim_{n\to\infty} \frac{Z(\lfloor sn\rfloor)}{\sigma
\sqrt{n}}=B(s),$$ where $\lfloor x\rfloor$ denotes the integer part of $x$ and $B(s)$ is a standard Brownian motion.

\[thm:A3\_b0\] Suppose $U_i=\{i-1,i,i+1\}$, $i\in\M$. Then with probability $1$ there exists the limit ${\bf x}=\lim_{t{\rightarrow}\infty}\xi(t)/t$, which takes a finite number of possible values with positive probabilities. The set of limiting configurations consists of those ${\bf x}=(x_1,\dots,x_{M})\in\R^M$ which simultaneously satisfy the following properties:

-   there exists an $\a>0$ such that $x_i\in\left\{0,\a/2,\a\right\}$ for all $i\in\M$; also $\sum_{i=1}^{M} x_i=1$;

-   if $x_i=0$, then $x_{i-1}>0$ or $x_{i+1}>0$, or both;

-   if $x_i=\a/2$, then $$(x_{j-3},x_{j-2},x_{j-1},x_{j},x_{j+1},x_{j+2})=(\a,0,\a/2,\a/2,0,
    \a),$$ where $j\in\{i,i+1\}$;

-   if $x_i=\a$, then $x_{i-1}=x_{i+1}=0$;

-   if $M=3K$ is divisible by $3$, then $$\min\{ x_{j},x_{j+3},x_{j+6},\dots,x_{j+3(K-1)}\}=0,$$ for $j=1$, $2$, $3$.

Moreover, 