Infinite excursions of router walks on regular trees

International audienceA router configuration on a graph contains in every vertex an infinite ordered sequence of routers, each is pointing to a neighbor of the vertex. After sampling a configuration according to some probability measure, a router walk is a deterministic process: at each step it chooses the next unused router in its current location, and uses it to jump to the neighboring vertex to which it points. Routers walks capture many aspects of the expected behavior of simple random walks. However, this similarity breaks down for the property of having an infinite excursion. In this paper we study that question for natural random configuration models on regular trees. Our results suggest that in this context the router model behaves like the simple random walk unless it is not "close to" the standard rotor-router model