Dead branches in the trace-reinforced ant process

In the trace-reinforced ant process of Kious, Mailler and Schapira, ants walk one after another from a nest $N$ to a food source $F$ on a finite graph, and each ant adds one unit of weight to every edge it has crossed. It was conjectured that the normalised weights converge to a deterministic limit, and that this limit is positive on every edge as soon as $N$ and $F$ are at distance at least two. We show that the second statement fails. If a part of the graph hangs by a single edge from a neighbour of $F$, on the side away from $N$, then the normalised weights of all its edges tend to zero almost surely; this happens in $2$-connected graphs with $N$ and $F$ at distance two. We also show that the mean-field vector field has no equilibrium with all coordinates positive when some neighbour of $F$ is hidden from $N$ by the other neighbours of $F$, so that on such graphs, some of which are $3$-connected, the normalised weights cannot converge to a positive limit. The equilibria are described through the voltage of the unit current from $N$ to $F$. A census of the $22{,}828$ cases on connected graphs with at most seven vertices supports a corrected form of the conjecture, in which the set of edges with limit zero is given by an explicit combinatorial rule. We also give the equilibrium on a graph made of any number of parallel paths.