$k\in\M$, there are no consecutive subsequences of the form $(0,0,\ast,0,0)$ where $\ast$ stands for any positive number; that is there is no $k$ such that v\_[k-2]{}(t\_j)=v\_[k-1]{}(t\_j)=0=v\_[k+1]{}(t\_j)=v\_[k+2]{}(t\_j) v\_[k]{}(t\_j)&gt;0.&

Propositions \[prop\_sym2\] and \[prop\_sym4\] imply that for some (random) $J$ large enough for all $j\ge J$ consecutive subsequences of zero (non-zero resp.) elements have length $\le 2$, and Proposition \[prop\_sym6\] says that two consecutive $0$s must be followed (preceded resp.) by a single non-zero element. Therefore, $(0,0,\ast,0,0)$ must be a part of a longer subsequence of form $(0,\ast,0,0,\ast,0,0,\ast,0)$. This, in turn, implies for the middle non-zero element located at $k$ that v\_k(t\_[j+1]{})={

[ll]{} v\_k(t\_j)+1, &,\
v\_k(t\_j), &,\
v\_k(t\_j)-1, &.

. Hence, by the properties of simple random walk, for some $j'>J$ we will have $v_k(t_{j'})=0$ (suppose that $j'$ is the first such time). On the other hand, by Proposition \[prop\_sym1\], $$v_{k-2}(t_{j'})=v_{k-1}(t_{j'})=v_{k+1}(t_{j'})=v_{k+2}(t_{j'})=0$$ as well. This yields a contradiction with the choice of $J$ (see Proposition \[prop\_sym4\]).

Let a *configuration of the potential* be a sequence $\bar v=(\bar v_1, \dots, \bar
v_M)$ where each $\bar v_i\in\{0,\ast\}$. Then we say that $v=\break(v_1,v_2,\dots,v_M)$ with the following property has type $\bar v$: v\_i=0& &|v\_i=0,\
v\_i&gt;0& &|v\_i=.

Propositions \[prop\_sym2\], \[prop\_sym4\], \[prop\_sym6\], and \[prop\_sym7\] rule out various types of configurations 