particular, if the maximum is unique, that is, $S_{\max}(s+1)=\{i\}$, then for all times $t\ge s$ this property will hold, and hence all the particles from now on will be adsorbed at $i$ only.

If, on the other hand, $|S_{\max}(s+1)|=2$, without loss of generality say $S_{\max}(s+1)=\{i,i+1\}$, then this property will be also preserved for all $t>s$ and each new particle will be adsorbed with probability $\frac 12$ at either $i$ or $i+1$.

Finally, if $|S_{\max}(s+1)|=3$, say $S_{\max}(s+1)=\{i,i+1,i+2\}$, then at time $s+2$ either $S_{\max}(s+2)=\{i,i+1,i+2\}$ if the adsorption occurred at $i+1$, or $S_{\max}(s+2)=\{i+1,i+2\}$ or $\{i,i+1\}$ otherwise. By iterating this argument we obtain that after a geometric number of times we will arrive at the situation where $|S_{\max}(t)|=2$, and then the process will follow the pattern described in the previous paragraph.

A similar simple argument shows that in the case of the asymmetric interaction only the outcome (\[sym\_inf\]) is possible.

[99]{} natexlab \#1[\#1]{} selectlanguage \#1

Bousch, T., and Mairesse, J. 2002. Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. *J. Amer. Math. Soc.*, [**15**]{}, 77–111.

Durrett, R., Kesten, H., and Limic, V. 2002. Once edge-reinforced random walk on a tree. *Probab. Theory Related Fields*, [**122**]{}, 567–592.

Evans, J. W. 1993. 